A006318 -id:A006318 - cp's OEIS frontend

A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 25, 9, 1, 1, 11, 41, 63, 41, 11, 1, 1, 13, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1, 1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1, 1, 21, 181, 833, 2241, 3653, 3653

Offset: 0

N. J. A. Sloane

In the Formula section, some contributors use T(n,k) = D(n-k, k) (for 0 <= k <= n), which is the triangular version of the square array (D(n,k): n,k >= 0). Conversely, D(n,k) = T(n+k,k) for n,k >= 0. - Petros Hadjicostas, Aug 05 2020
Also called the tribonacci triangle [Alladi and Hoggatt (1977)]. - N. J. A. Sloane, Mar 23 2014
D(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), (1,1). - Joerg Arndt, Jul 01 2011 [Corrected by N. J. A. Sloane, May 30 2020]
Or, triangle read by rows of coefficients of polynomials P[n](x) defined by P[0] = 1, P[1] = x+1; for n >= 2, P[n] = (x+1)*P[n-1] + x*P[n-2].
D(n, k) is the number of k-matchings of a comb-like graph with n+k teeth. Example: D(1, 3) = 7 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has seven 3-matchings: four triples of three teeth and the three triples {Aa, Bb, CD}, {Aa, Dd, BC}, {Cc, Dd, AB}. Also D(3, 1)=7, the 1-matchings of the same graph being the seven edges: {AB}, {BC}, {CD}, {Aa}, {Bb}, {Cc}, {Dd}. - Emeric Deutsch, Jul 01 2002
Sum of n-th antidiagonal of the array D is A000129(n+1). - Reinhard Zumkeller, Dec 03 2004 [Edited by Petros Hadjicostas, Aug 05 2020 so that the counting of antidiagonals of D starts at n = 0. That is, the sum of the terms in the n-th row of the triangles T is  A000129(n+1).]
The A-sequence for this Riordan type triangle (see one of Paul Barry's comments under Formula) is A112478 and the Z-sequence the trivial: {1, 0, 0, 0, ...}. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. This leads to the recurrence for the triangle given below. - Wolfdieter Lang, Jan 21 2008
The triangle or chess sums, see A180662 for their definitions, link the Delannoy numbers with twelve different sequences, see the crossrefs. All sums come in pairs due to the symmetrical nature of this triangle. The knight sums Kn14 and Kn15 have been added. It is remarkable that all knight sums are related to the tribonacci numbers, that is, A000073 and A001590, but none of the others. - Johannes W. Meijer, Sep 22 2010
This sequence, A008288, is jointly generated with A035607 as an array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 09 2012
Row n, for n > 0, of Roger L. Bagula's triangle in the Example section shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the numerator of the n-th convergent of the continued fraction [k, k, k, ...], where k = sqrt(x) + 1/sqrt(x); see A230000. - Clark Kimberling, Nov 13 2013
In an n-dimensional hypercube lattice, D(n,k) gives the number of nodes situated at a Minkowski (Manhattan) distance of k from a given node. In cellular automata theory, the cells at Manhattan distance k are called the von Neumann neighborhood of radius k. For k=1, see A005843. - Dmitry Zaitsev, Dec 10 2015
These numbers appear as the coefficients of series relating spherical and bispherical harmonics, in the solutions of Laplace's equation in 3D. [Majic 2019, Eq. 22] - Matt Majic, Nov 24 2019
From Peter Bala, Feb 19 2020: (Start)
The following remarks assume an offset of 1 in the row and column indices of the triangle.
The sequence of row polynomials T(n,x), beginning with T(1,x) = x, T(2,x) = x + x^2, T(3,x) = x + 3*x^2 + x^3, ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd(T(n,x), T(m,x)) = T(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the sequence (T(n,x): n >= 1) is a divisibility sequence in the polynomial ring Z[x]; that is, if n divides m then T(n,x) divides T(m,x) in Z[x].
Let S(x) = 1 + 2*x + 6*x^2 + 22*x^3 + ... denote the o.g.f. for the large Schröder numbers A006318. The power series (x*S(x))^n, n = 2, 3, 4, ..., can be expressed as a linear combination with polynomial coefficients of S(x) and 1: (x*S(x))^n = T(n-1,-x) - T(n,-x)*S(x). The result can be extended to negative integer n if we define T(0,x) = 0 and T(-n,x) = (-1)^(n+1) * T(n,x)/x^n. Cf. A115139.
[In the previous two paragraphs, D(n,x) was replaced with T(n,x) because the contributor is referring to the rows of the triangle T(n,k), not the rows of the array D(n,k). - Petros Hadjicostas, Aug 05 2020] (End)
Named after the French amateur mathematician Henri-Auguste Delannoy (1833-1915). - Amiram Eldar, Apr 15 2021
D(i,j) = D(j,i). With this and Dmitry Zaitsev's Dec 10 2015 comment, D(i,j) can be considered the number of points at L1 distance <= i in Z^j or the number of points at L1 distance <= j in Z^i from any given point. The rows and columns of D(i,j) are the crystal ball sequences on cubic lattices. See the first example below. The n-th term in the k-th crystal ball sequence can be considered the number of points at distance <= n from any point in a k-dimensional cubic lattice, or the number of points at distance <= k from any point in an n-dimensional cubic lattice. - Shel Kaphan, Jan 01 2023 and Jan 07 2023
Dimensions of hom spaces Hom(R^{(i)}, R^{(j)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - Noah Snyder, Mar 22 2023

			The square array D(i,j) (i >= 0, j >= 0) begins:
  1, 1,  1,   1,   1,   1,    1,    1,    1,    1, ... = A000012
  1, 3,  5,   7,   9,  11,   13,   15,   17,   19, ... = A005408
  1, 5, 13,  25,  41,  61,   85,  113,  145,  181, ... = A001844
  1, 7, 25,  63, 129, 231,  377,  575,  833, 1159, ... = A001845
  1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, ... = A001846
  ...
For D(2,5) = 61, which is seen above in the row labeled A001844, we calculate the sum (9 + 11 + 41) of the 3 nearest terms above and/or to the left. - _Peter Munn_, Jan 01 2023
D(2,5) = 61 can also be obtained from the row labeled A005408 using a recurrence mentioned in the formula section:  D(2,5) = D(1,5) + 2*Sum_{k=0..4} D(1,k), so D(2,5) = 11 + 2*(1+3+5+7+9) = 11 + 2*25. - _Shel Kaphan_, Jan 01 2023
As a triangular array (on its side) this begins:
   0,   0,   0,   0,   1,   0,  11,   0, ...
   0,   0,   0,   1,   0,   9,   0,  61, ...
   0,   0,   1,   0,   7,   0,  41,   0, ...
   0,   1,   0,   5,   0,  25,   0, 129, ...
   1,   0,   3,   0,  13,   0,  63,   0, ...
   0,   1,   0,   5,   0,  25,   0, 129, ...
   0,   0,   1,   0,   7,   0,  41,   0, ...
   0,   0,   0,   1,   0,   9,   0,  61, ...
   0,   0,   0,   0,   1,   0,  11,   0, ...
   [Edited by _Shel Kaphan_, Jan 01 2023]
From _Roger L. Bagula_, Dec 09 2008: (Start)
As a triangle T(n,k) (with rows n >= 0 and columns k = 0..n), this begins:
   1;
   1,  1;
   1,  3,   1;
   1,  5,   5,   1;
   1,  7,  13,   7,    1;
   1,  9,  25,  25,    9,    1;
   1, 11,  41,  63,   41,   11,    1;
   1, 13,  61, 129,  129,   61,   13,   1;
   1, 15,  85, 231,  321,  231,   85,  15,   1;
   1, 17, 113, 377,  681,  681,  377, 113,  17,  1;
   1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1;
   ... (End)
Triangle T(n,k) recurrence: 63 = T(6,3) = 25 + 13 + 25 = T(5,2) + T(4,2) + T(5,3).
Triangle T(n,k) recurrence with A-sequence A112478: 63 = T(6,3) = 1*25 + 2*25 - 2*9 + 6*1 (T entries from row n = 5 only). [Here the formula T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+j) is used with n = 6 and k = 3; i.e., T(6,3) = Sum_{j=0..3} A111478(j) * T(5, 2+j). - _Petros Hadjicostas_, Aug 05 2020]
From _Philippe Deléham_, Mar 29 2012: (Start)
Subtriangle of the triangle given by (1, 0, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:
   1;
   1,  0;
   1,  1,  0;
   1,  3,  1,  0;
   1,  5,  5,  1,  0;
   1,  7, 13,  7,  1,  0;
   1,  9, 25, 25,  9,  1, 0;
   1, 11, 41, 63, 41, 11, 1, 0;
   ...
Subtriangle of the triangle given by (0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:
   1;
   0, 1;
   0, 1,  1;
   0, 1,  3,  1;
   0, 1,  5,  5,  1;
   0, 1,  7, 13,  7,  1;
   0, 1,  9, 25, 25,  9,  1;
   0, 1, 11, 41, 63, 41, 11, 1;
   ... (End)

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Sums of antidiagonals: A000129 (Pell numbers).
Main diagonal: A001850 (central Delannoy numbers), which has further information and references.
A002002, A026002, and A190666 are +-k-diagonals for k=1, 2, 3 resp. - Shel Kaphan, Jan 01 2023
Rows 0..10: A000012, A005408, A001844, A001845, A001846, A001847, A001848, A001849, A008417, A008419, A008421.
See also A027618.
Cf. A059446.
Has same main diagonal as A064861. Different from A100936.
Cf. A101164, A101167, A128966, A131887, A131935.
Cf. A035607, A108625, A142979, A142992, A143007.
Read mod small primes: A211312, A211313, A211314, A211315.
Triangle sums (see the comments): A000129 (Row1); A056594 (Row2); A000073 (Kn11 & Kn21); A089068 (Kn12 & Kn22); A180668 (Kn13 & Kn23); A180669 (Kn14 & Kn24); A180670 (Kn15 & Kn25); A099463 (Kn3 & Kn4); A116404 (Fi1 & Fi2); A006498 (Ca1 & Ca2); A006498(3*n) (Ca3 & Ca4); A079972 (Gi1 & Gi2); A079972(4*n) (Gi3 & Gi4); A079973(3*n) (Ze1 & Ze2); A079973(2*n) (Ze3 & Ze4).
Cf. A102413, A128966. (D(n,1)) = A005843. Cf. A115139.

a008288 n k = a008288_tabl !! n !! k
a008288_row n = a008288_tabl !! n
a008288_tabl = map fst $ iterate
    (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                       zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Jul 21 2013

A008288 := proc(n, k) option remember; if k = 0 then 1 elif n=k then 1 else procname(n-1, k-1) + procname(n-2, k-1) + procname(n-1, k) end if; end proc: seq(seq(A008288(n,k),k=0..n), n=0..10); # triangular indices n and k
P[0]:=1; P[1]:=x+1; for n from 2 to 12 do P[n]:=expand((x+1)*P[n-1]+x*P[n-2]); lprint(P[n]); lprint(seriestolist(series(P[n],x,200))); end do:

(* Next, A008288 jointly generated with A035607 *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A008288 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A035607 *)
(* Clark Kimberling, Mar 09 2012 *)
d[n_, k_] := Binomial[n+k, k]*Hypergeometric2F1[-k, -n, -n-k, -1]; A008288 = Flatten[Table[d[n-k, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Apr 05 2012, after 3rd formula *)

from functools import cache
@cache
def delannoy_row(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [1, 1]
    rov = delannoy_row(n - 2)
    row = delannoy_row(n - 1) + [1]
    for k in range(n - 1, 0, -1):
        row[k] += row[k - 1] + rov[k - 1]
    return row
for n in range(10): print(delannoy_row(n))  # Peter Luschny, Jul 30 2023

for k in range(8):  # seen as an array, read row by row
    a = lambda n: hypergeometric([-n, -k], [1], 2)
    print([simplify(a(n)) for n in range(11)]) # Peter Luschny, Nov 19 2014

D(n, 0) = 1 = D(0, n) for n >= 0; D(n, k) = D(n, k-1) + D(n-1, k-1) + D(n-1, k).
Bivariate o.g.f.: Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1 - x - y - x*y).
D(n, k) = Sum_{d = 0..min(n,k)} binomial(k, d)*binomial(n+k-d, k) = Sum_{d=0..min(n,k)} 2^d*binomial(n, d)*binomial(k, d). [Edited by Petros Hadjicostas, Aug 05 2020]
Seen as a triangle read by rows: T(n, 0) = T(n, n) = 1 for n >= 0 and T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), 0 < k < n and n > 1. - Reinhard Zumkeller, Dec 03 2004
Read as a number triangle, this is the Riordan array (1/(1-x), x(1+x)/(1-x)) with T(n, k) = Sum_{j=0..n-k} C(n-k, j) * C(k, j) * 2^j. - Paul Barry, Jul 18 2005
T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k). - Paul Barry, May 21 2006
Let y^k(n) be the number of Khalimsky-continuous functions f from [0,n-1] to Z such that f(0) = 0 and f(n-1) = k. Then y^k(n) = D(i,j) for i = (1/2)*(n-1-k) and j = (1/2)*(n-1+k) where n-1+k belongs to 2Z. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
Recurrence for triangle from A-sequence (see the Wolfdieter Lang comment above): T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+j), n >= 1, k >= 1. [For k > n, the sum is empty, in which case T(n,k) = 0.]
From Peter Bala, Jul 17 2008: (Start)
The n-th row of the square array is the crystal ball sequence for the product lattice A_1 x ... x A_1 (n copies). A035607 is the table of the associated coordination sequences for these lattices.
The polynomial p_n(x) := Sum {k = 0..n} 2^k * C(n,k) * C(x,k) = Sum_{k = 0..n} C(n,k) * C(x+k,n), whose values [p_n(0), p_n(1), p_n(2), ... ] give the n-th row of the square array, is the Ehrhart polynomial of the n-dimensional cross polytope (the hyperoctahedron) [Bump et al. (2000), Theorem 6].
The first few values are p_0(x) = 1, p_1(x) = 2*x + 1, p_2(x) = 2*x^2 + 2*x + 1 and p_3(x) = (4*x^3 + 6*x^2 + 8*x + 3)/3.
The reciprocity law p_n(m) = p_m(n) reflects the symmetry of the table.
The polynomial p_n(x) is the unique polynomial solution of the difference equation (x+1)*f(x+1) - x*f(x-1) = (2*n+1)*f(x), normalized so that f(0) = 1.
These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane; that is, the polynomials p_n(x-1), n = 1,2,3,..., satisfy a Riemann hypothesis [Bump et al. (2000), Theorem 4]. The o.g.f. for the p_n(x) is (1 + t)^x/(1 - t)^(x + 1) = 1 + (2*x + 1)*t + (2*x^2 + 2*x + 1)*t^2 + ... .
The square array of Delannoy numbers has a close connection with the constant log(2). The entries in the n-th row of the array occur in the series acceleration formula log(2) = (1 - 1/2 + 1/3 - ... + (-1)^(n+1)/n) + (-1)^n * Sum_{k>=1} (-1)^(k+1)/(k*D(n,k-1)*D(n,k)). [T(n,k) was replaced with D(n,k) in the formula to agree with the beginning of the paragraph. - Petros Hadjicostas, Aug 05 2020]
For example, the fourth row of the table (n = 3) gives the series log(2) = 1 - 1/2 + 1/3 - 1/(1*1*7) + 1/(2*7*25) - 1/(3*25*63) + 1/(4*63*129) - ... . See A142979 for further details.
Also the main diagonal entries (the central Delannoy numbers) give the series acceleration formula Sum_{n>=1} 1/(n*D(n-1,n-1)*D(n,n)) = (1/2)*log(2), a result due to Burnside. [T(n,n) was replaced here with D(n,n) to agree with the previous paragraphs. - Petros Hadjicostas, Aug 05 2020]
Similar relations hold between log(2) and the crystal ball sequences of the C_n lattices A142992. For corresponding results for the constants zeta(2) and zeta(3), involving the crystal ball sequences for root lattices of type A_n and A_n x A_n, see A108625 and A143007 respectively. (End)
From Peter Bala, Oct 28 2008: (Start)
Hilbert transform of Pascal's triangle A007318 (see A145905 for the definition of this term).
D(n+a,n) = P_n(a,0;3) for all integer a such that a >= -n, where P_n(a,0;x) is the Jacobi polynomial with parameters (a,0) [Hetyei]. The related formula A(n,k) = P_k(0,n-k;3) defines the table of asymmetric Delannoy numbers, essentially A049600. (End)
Seen as a triangle read by rows: T(n, k) = Hyper2F1([k-n, -k], [1], 2). - Peter Luschny, Aug 02 2014, Oct 13 2024.
From Peter Bala, Jun 25 2015: (Start)
O.g.f. for triangle T(n,k): A(z,t) = 1/(1 - (1 + t)*z - t*z^2) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ....
1 + z*d/dz(A(z,t))/A(z,t) is the o.g.f. for A102413. (End)
E.g.f. for the n-th subdiagonal of T(n,k), n >= 0, equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(2*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 4*x + 4*x^2/2) = 1 + 5*x + 13*x^2/2! + 25*x^3/3! + 41*x^4/4! + 61*x^5/5! + .... - Peter Bala, Mar 05 2017 [The n-th subdiagonal of triangle T(n,k) is the n-th row of array D(n,k).]
Let a_i(n) be multiplicative with a_i(p^e) = D(i, e), p prime and e >= 0, then Sum_{n > 0} a_i(n)/n^s = (zeta(s))^(2*i+1)/(zeta(2*s))^i for i >= 0. - Werner Schulte, Feb 14 2018
Seen as a triangle read by rows: T(n,k) = Sum_{i=0..k} binomial(n-i, i) * binomial(n-2*i, k-i) for 0 <= k <= n. - Werner Schulte, Jan 09 2019
Univariate generating function: Sum_{k >= 0} D(n,k)*z^k = (1 + z)^n/(1 - z)^(n+1). [Dziemianczuk (2013), Eq. 5.3] - Matt Majic, Nov 24 2019
(n+1)*D(n+1,k) = (2*k+1)*D(n,k) + n*D(n-1,k). [Majic (2019), Eq. 22] - Matt Majic, Nov 24 2019
For i, j >= 1, D(i,j) = D(i,j-1) + 2*Sum_{k=0..i-1} D(k,j-1), or, because D(i,j) = D(j,i), D(i,j) = D(i-1,j) + 2*Sum_{k=0..j-1} D(i-1,k). - Shel Kaphan, Jan 01 2023
Sum_{k=0..n} T(n,k)^2 = A026933(n). - R. J. Mathar, Nov 07 2023
Let S(x) = (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x) denote the g.f. of the sequence of large Schröder numbers A006318. Read as a lower triangular array, the signed n-th row polynomial R(n, -x) = 1/sqrt(1 - 6*x + x^2) *( 1/S(x)^(n+1) + (x*S(x))^(n+1) ). For example, R(4, -x) = 1 - 7*x + 13*x^2 - 7*x^3 + x^4 = 1/sqrt(1 - 6*x + x^2) * ( 1/S(x)^5 + (x*S(x))^5 ). Cf. A102413. - Peter Bala, Aug 01 2024

Expanded description from Clark Kimberling, Jun 15 1997
Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001
Changed the notation to make the formulas more precise. - N. J. A. Sloane, Jul 01 2002

A027307 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

Original entry on oeis.org

1, 2, 10, 66, 498, 4066, 34970, 312066, 2862562, 26824386, 255680170, 2471150402, 24161357010, 238552980386, 2375085745978, 23818652359682, 240382621607874, 2439561132029314, 24881261270812490, 254892699352950850

Offset: 0

Emeric Deutsch

These are the 3-Schroeder numbers according to Yang-Jiang (2021). - N. J. A. Sloane, Mar 28 2021
Equals row sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna, Mar 30 2005
a(n) counts ordered complete ternary trees with 2*n + 1 leaves, where the internal vertices come in two colors and such that each vertex and its rightmost child have different colors. See [Drake, Example 1.6.9]. An example is given below. - Peter Bala, Sep 29 2011
a(n) for n >= 1 is the number of compact coalescent histories for matching lodgepole gene trees and species trees with n cherries and 2n+1 leaves. - Noah A Rosenberg, Jun 21 2022
a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... union S_{2*n} intersect S_{2*n+1}”, where n union and n intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{2*n+1} are sets. - Alexander Burstein, Nov 22 2023

			a(2) = 10. Internal vertices colored either b(lack) or w(hite); 5 uncolored leaf vertices shown as o.
........b...........b.............w...........w.....
......./|\........./|\.........../|\........./|\....
....../.|.\......./.|.\........./.|.\......./.|.\...
.....b..o..o.....o..b..o.......w..o..o.....o..w..o..
..../|\............/|\......../|\............/|\....
.../.|.\........../.|.\....../.|.\........../.|.\...
..o..o..o........o..o..o....o..o..o........o..o..o..
....................................................
........b...........b.............w...........w.....
......./|\........./|\.........../|\........./|\....
....../.|.\......./.|.\........./.|.\......./.|.\...
.....w..o..o.....o..w..o.......b..o..o.....o..b..o..
..../|\............/|\......../|\............/|\....
.../.|.\........../.|.\....../.|.\........../.|.\...
..o..o..o........o..o..o....o..o..o........o..o..o..
....................................................
........b...........w..........
......./|\........./|\.........
....../.|.\......./.|.\........
.....o..o..w.....o..o..b.......
........../|\........./|\......
........./.|.\......./.|.\.....
........o..o..o.....o..o..o....
...............................
From _Alexander Burstein_, Feb 14 2025: (Start)
a(2) = 10 as the maximum number of distinct sets obtained as complete parenthesizations of S_1 u(nion) S_2 (i)n(tersect) S_3 u(nion) S_4 (i)n(tersect) S_5:
S_1 u (S_2 n (S_3 u (S_4 n S_5))),
S_1 u (S_2 n ((S_3 u S_4) n S_5)) = S_1 u ((S_2 n (S_3 u S_4)) n S_5),
S_1 u ((S_2 n S_3) u (S_4 n S_5)) = (S_1 u (S_2 n S_3)) u (S_4 n S_5),
S_1 u (((S_2 n S_3) u S_4) n S_5),
(S_1 u S_2) n (S_3 u (S_4 n S_5)),
(S_1 u S_2) n ((S_3 u S_4) n S_5) = ((S_1 u S_2) n (S_3 u S_4)) n S_5,
((S_1 u S_2) n S_3) u (S_4 n S_5),
(S_1 u (S_2 n (S_3 u S_4))) n S_5,
(S_1 u ((S_2 n S_3) u S_4)) n S_5 = ((S_1 u (S_2 n S_3)) u S_4) n S_5,
(((S_1 u S_2) n S_3) u S_4) n S_5. (End)

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Gi-Sang Cheon, S.-T. Jin and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
F. Disanto and N. A. Rosenberg, Enumeration of compact coalescent histories for matching gene trees and species trees, J. Math. Biol 78 (2019), 155-188.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.9), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Zhen-ji Tian and Shu-ping Dou, Enumerations of 3 di-sk trees, J. Lanzhou Univ. Tech. (2024) Vol. 50, No. 6, 144-149. See p. 146.
J. Winter, M. M. Bonsangue and J. J. M. M. Rutten, Context-free coalgebras, 2013.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 11.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A104978. A196201.
The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021
Apart from first term, this is 2*A034015. - N. J. A. Sloane, Mar 28 2021

a[n_] := ((n+1)*(2n)!*Hypergeometric2F1[-n, 2n+1, n+2, -1]) / (n+1)!^2;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 14 2011, after Pari *)
a[n_] := If[n == 0, 1, 2*Hypergeometric2F1[1 - n, -2 n, 2, 2]];
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Nov 08 2021 *)

a(n)=if(n<1,n==0,sum(i=0,n-1,2^(i+1)*binomial(2*n,i)*binomial(n,i+1))/n)

a(n)=sum(k=0,n,binomial(2*n+k,n+2*k)*binomial(n+2*k,k)/(n+k+1)) \\ Paul D. Hanna

a(n)=sum(k=0,n, binomial(n,k)*binomial(2*n+k+1,n)/(2*n+k+1) ) /* Michael Somos, May 23 2005 */

G.f.: (2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3.
a(n) = (1/n) * Sum_{i=0..n-1} 2^(i+1)*binomial(2*n, i)*binomial(n, i+1), n>0.
a(n) = 2*A034015(n-1), n>0.
a(n) = Sum_{k=0..n} C(2*n+k, n+2*k)*C(n+2*k, k)/(n+k+1). - Paul D. Hanna, Mar 30 2005
Given g.f. A(x), y=A(x)x satisfies 0=f(x, y) where f(x, y)=x(x-y)+(x+y)y^2 . - Michael Somos, May 23 2005
Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A085403(k)x^k).
G.f. A(x) satisfies A(x)=A006318(x*A(x)). - Vladimir Kruchinin, Apr 18 2011
The function B(x) = x*A(x^2) satisfies B(x) = x+x*B(x)^2+B(x)^3 and hence B(x) = compositional inverse of x*(1-x^2)/(1+x^2) = x+2*x^3+10*x^5+66*x^7+.... Let f(x) = (1+x^2)^2/(1-4*x^2+x^4) and let D be the operator f(x)*d/dx. Then a(n) equals 1/(2*n+1)!*D^(2*n)(f(x)) evaluated at x = 0. For a refinement of this sequence see A196201. - Peter Bala, Sep 29 2011
D-finite with recurrence: 2*n*(2*n+1)*a(n) = (46*n^2-49*n+12)*a(n-1) - 3*(6*n^2-26*n+27)*a(n-2) - (n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(50+30*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012. Equivalently, a(n) ~ phi^(5*n + 1) / (2 * 5^(1/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
a(n) = 2*hypergeom([1 - n, -2*n], [2], 2) for n >= 1. - Peter Luschny, Nov 08 2021
From Peter Bala, Jun 16 2023: (Start)
P-recursive: n*(2*n + 1)*(5*n - 7)*a(n) = (110*n^3 - 264*n^2 + 181*n - 36)*a(n-1) + (n - 2)*(2*n - 3)*(5*n - 2)*a(n-2) with a(0) = 1 and a(1) = 2.
The g.f. A(x) = 1 + 2*x + 10*x^2 + 66*x^3 + ... satisfies A(x)^2 = (1/x) * the series reversion of x*((1 - x)/(1 + x))^2.
Define b(n) = [x^(2*n)] ( (1 + x)/(1 - x) )^n = (1/2) * [x^n] ((1 + x)/(1 - x))^(2*n) = A103885(n). Then A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ). (End)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 09 2023

A027941 a(n) = Fibonacci(2*n + 1) - 1.

Original entry on oeis.org

0, 1, 4, 12, 33, 88, 232, 609, 1596, 4180, 10945, 28656, 75024, 196417, 514228, 1346268, 3524577, 9227464, 24157816, 63245985, 165580140, 433494436, 1134903169, 2971215072, 7778742048, 20365011073, 53316291172, 139583862444, 365435296161, 956722026040

Offset: 0

Clark Kimberling

Also T(2n+1,n+1), T given by A027935. Also first row of Inverse Stolarsky array.
Third diagonal of array defined by T(i, 1)=T(1, j)=1, T(i, j)=Max(T(i-1, j)+T(i-1, j-1); T(i-1, j-1)+T(i, j-1)). - Benoit Cloitre, Aug 05 2003
Number of Schroeder paths of length 2(n+1) having exactly one up step starting at an even height (a Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis). Schroeder paths are counted by the large Schroeder numbers (A006318). Example: a(1)=4 because among the six Schroeder paths of length 4 only the paths (U)HD, (U)UDD, H(U)D, (U)DH have exactly one U step that starts at an even height (shown between parentheses). - Emeric Deutsch, Dec 19 2004
Also: smallest number not writeable as the sum of fewer than n positive Fibonacci numbers. E.g., a(5)=88 because it is the smallest number that needs at least 5 Fibonacci numbers: 88 = 55 + 21 + 8 + 3 + 1. - Johan Claes, Apr 19 2005 [corrected for offset and clarification by Mike Speciner, Sep 19 2023] In general, a(n) is the sum of n positive Fibonacci numbers as a(n) = Sum_{i=1..n} A000045(2*i). See A001076 when negative Fibonacci numbers can be included in the sum. - Mike Speciner, Sep 24 2023
Except for first term, numbers a(n) that set a new record in the number of Fibonacci numbers needed to sum up to n. Position of records in sequence A007895. - Ralf Stephan, May 15 2005
Successive extremal petal bends beta(n) = a(n-2). See the Ring Lemma of Rodin and Sullivan in K. Stephenson, Introduction to Circle Packing (Cambridge U. P., 2005), pp. 73-74 and 318-321. - David W. Cantrell (DWCantrell(AT)sigmaxi.net)
a(n+1)= AAB^(n)(1), n>=1, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 4=`110`, 12=`1100`, 33=`11000`, 88=`110000`, ..., in Wythoff code. AA(1)=1=a(1) but for uniqueness reason 1=A(1) in Wythoff code. - N. J. A. Sloane, Jun 29 2008
Start with n. Each n generates a sublist {n-1,n-1,n-2,..,1}. Each element of each sublist also generates a sublist. Add numbers in all terms. For example, 3->{2,2,1} and both 2->{1,1}, so a(3) = 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 = 12. - Jon Perry, Sep 01 2012
For n>0: smallest number such that the inner product of Zeckendorf binary representation and its reverse equals n: A216176(a(n)) = n, see also A189920. - Reinhard Zumkeller, Mar 10 2013
Also, numbers m such that 5*m*(m+2)+1 is a square. - Bruno Berselli, May 19 2014
Also, number of nonempty submultisets of multisets of weight n that span an initial interval of integers (see 2nd example). - Gus Wiseman, Feb 10 2015
From Robert K. Moniot, Oct 04 2020: (Start)
Including a(-1):=0, consecutive terms (a(n-1),a(n))=(u,v) or (v,u) give all points on the hyperbola u^2-u+v^2-v-4*u*v=0 with both coordinates nonnegative integers.  Note that this follows from identifying (1,u+1,v+1) with the Markov triple (1,Fibonacci(2n-1),Fibonacci(2n+1)).  See A001519 (comments by Robert G. Wilson, Oct 05 2005, and Wolfdieter Lang, Jan 30 2015).
Let T(n) denote the n-th triangular number. If i, j are any two successive elements of the above sequence then (T(i-1) + T(j-1))/T(i+j-1) = 3/5. (End)

			a(5) = 88 = 2*33 + 12 + 4 + 1 + 5. a(6) = 232 = 2*88 + 33 + 12 + 4 + 1 + 6. - _Jon Perry_, Sep 01 2012
a(4) = 33 counts all nonempty submultisets of the last row: [1][2][3][4], [11][12][13][14][22][23][24][33][34], [111][112][113][122][123][124][133][134][222][223][233][234], [1111][1112][1122][1123][1222][1223][1233][1234]. - _Gus Wiseman_, Feb 10 2015

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 12.

T. D. Noe, Table of n, a(n) for n = 0..200
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Russ Euler and Jawad Sadek, Problem B-912, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 39, No. 1 (2001), p. 85; From a Product to a Sum, Solution to Problem B-912 by Charles K. Cook, ibid., Vol. 39, No. 5 (2001), pp. 468-469.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 7.
Clark Kimberling, Interspersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 60 (doubled).
Luis A. Medina and Armin Straub, On multiple and infinite log-concavity, Annals of Combinatorics, Vol. 20, No. 1 (2016), pp. 125-138; arXiv preprint, arXiv:1405.1765 [math.CO], 2014; preprint, 2014.
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (2nd line of Table 1 is 3*a(n-2)).
László Németh, Pascal pyramid in the space H^2×R, arXiv:1701.06022 [math.CO], 2016. See bn in Table 1 p.10.
N. J. A. Sloane, Classic Sequences.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000045, A035507, A001906, A006318, A000032.
Related to partial sums of Fibonacci(k*n) over n: A000071, A099919, A058038, A138134, A053606; this sequence is the case k=2.
Cf. A212336 for more sequences with g.f. of the type 1/(1 - k*x + k*x^2 - x^3).
Cf. A000225 (sublist connection).
Cf. A258993 (row sums, n > 0), A000967.

a027941 = (subtract 1) . a000045 . (+ 1) . (* 2)
-- Reinhard Zumkeller, Mar 10 2013

[Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

with(combinat): seq(fibonacci(2*n+1)-1,n=1..27); # Emeric Deutsch, Dec 19 2004
a:=n->sum(binomial(n+k+1,2*k), k=0..n): seq(a(n), n=-1..26); # Zerinvary Lajos, Oct 02 2007

Table[Fibonacci[2*n+1]-1,{n,0,17}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)
LinearRecurrence[{4,-4,1},{0,1,4},40] (* Harvey P. Dale, Aug 17 2021 *)

a(n):=sum(binomial(n+1,k+1)*fib(k),k,0,n); /* Vladimir Kruchinin, Oct 14 2016 */

a(n)=fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Nov 20 2012

concat(0, Vec(x/((1-x)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Jun 03 2016

a(n) = Sum_{i=1..n} binomial(n+i, n-i). - Benoit Cloitre, Oct 15 2002
G.f.: Sum_{k>=1} x^k/(1-x)^(2*k+1). - Benoit Cloitre, Apr 21 2003
a(n) = Sum_{k=1..n} F(2*k), i.e., partial sums of A001906. - Benoit Cloitre, Oct 27 2003
a(n) = Sum_{k=0..n-1} U(k, 3/2) = Sum_{k=0..n-1} S(k, 3), with S(k, 3) = A001906(k+1). - Paul Barry, Nov 14 2003
G.f.: x/((1-x)*(1-3*x+x^2)) = x/(1-4*x+4*x^2-x^3).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) with n>=2, a(-1)=0, a(0)=0, a(1)=1.
a(n) = 3*a(n-1) - a(n-2) + 1 with n>=1, a(-1)=0, a(0)=0.
a(n) = Sum_{k=1..n} F(k)*L(k), where L(k) = Lucas(k) = A000032(k) = F(k-1) + F(k+1). - Alexander Adamchuk, May 18 2007
a(n) = 2*a(n-1) + (Sum_{k=1..n-2} a(k)) + n. - Jon Perry, Sep 01 2012
Sum {n >= 1} 1/a(n) = 3 - phi, where phi = 1/2*(1 + sqrt(5)) is the golden ratio. The ratio of adjacent terms r(n) := a(n)/a(n-1) satisfies the recurrence r(n+1) = (4*r(n) - 1)/(r(n) + 1) for n >= 2. - Peter Bala, Dec 05 2013
a(n) = S(n, 3) - S(n-1, 3) - 1, n >= 0, with Chebyshev's S-polynomials (see A049310), where S(-1, x) = 0. - Wolfdieter Lang, Aug 28 2014
a(n) = -1 + (2^(-1-n)*((3-sqrt(5))^n*(-1+sqrt(5)) + (1+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5). - Colin Barker, Jun 03 2016
E.g.f.: (sqrt(5)*sinh(sqrt(5)*x/2) + 5*cosh(sqrt(5)*x/2))*exp(3*x/2)/5 - exp(x). - Ilya Gutkovskiy, Jun 03 2016
a(n) = Sum_{k=0..n} binomial(n+1,k+1)*Fibonacci(k). - Vladimir Kruchinin, Oct 14 2016
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} C(k+i+1,k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n)*a(n-2) = a(n-1)*(a(n-1) - 1) for n>1. - Robert K. Moniot, Aug 23 2020
a(n) = Sum_{k=1..n} C(2*n-k,k). - Wesley Ivan Hurt, Dec 22 2020
a(n) = Sum_{k = 1..2*n+2} (-1)^k*Fibonacci(k). - Peter Bala, Nov 14 2021
a(n) = (2*cosh((1 + 2*n)*arccsch(2)))/sqrt(5) - 1. - Peter Luschny, Nov 21 2021
a(n) = F(n + (n mod 2)) * L(n+1 - (n mod 2)), where L(n) = A000032(n) and F(n) = A000045(n) (Euler and Sadek, 2001). - Amiram Eldar, Jan 13 2022

More terms from James Sellers, Sep 08 2000

Paul Barry's Nov 14 2003 formula, recurrences and g.f. corrected for offset 0 and index link for Chebyshev polynomials added by Wolfdieter Lang, Aug 28 2014

A025227 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 3.

Original entry on oeis.org

0, 1, 2, 4, 12, 40, 144, 544, 2128, 8544, 35008, 145792, 615296, 2625792, 11311616, 49124352, 214838528, 945350144, 4182412288, 18593224704, 83015133184, 372090122240, 1673660915712, 7552262979584, 34178799378432, 155096251351040, 705533929816064

Offset: 0

Clark Kimberling

Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jul 27 2003
a(n) is the number of royal paths (A006318) from (0,0) to (n-1,n-1) such that every northeast (diagonal) step is either immediately followed by a north step or ends the path. For example a(3)=4 counts EDN, EENN, END, ENEN (E=east, D=diagonal, N=north). - David Callan, Jul 03 2006
From David Callan, Sep 25 2006: (Start)
a(n) is the number of ordered trees with n leaves in which (i) every node (= non-root non-leaf vertex) has at least 2 children and (ii) each leaf is either the leftmost or rightmost child of its parent. For example, a(3)=4 counts
              |
     /\      / \
      /\       /\
and their mirror images. (End)
From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010: (Start)
a(n+1), n >= 0, is also the number of Rota-Baxter words in one idempotent generator x and one operator of arity n.
Alternatively, a(n+1) is the number of ways of adding pairs of parentheses to a string of n x's (the number m of parentheses pairs necessarily satisfies m <= n <= 2m+1 for a nonzero count), such that no two pairs of parentheses are immediately nested and no two x's remain adjacent. (End)
a(n) is the number of colored binary trees on n-1 vertices where leaves have 2 possible colors and internal nodes have 1 color. - Alexander Burstein, Mar 07 2020

			For n=2, a(3) = 4 has the following words: x(x), (x)x, (x(x)), ((x)x) corresponding to A(1,2)=2, and A(2,2)=2. - William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010

L. Guo and W. Sit, Enumeration of Rota-Baxter Words (extended abstract), ISSAC 2006 Proceedings, 123-131. [From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010]
L. Guo and W. Sit, Enumeration of Rota-Baxter Words, to appear in Mathematics in Computer Science, Special Issue on AADIOS special session, ACA, 2009. [From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010]

Michael De Vlieger, Table of n, a(n) for n = 0..1470
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 22.
Maciej Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
Li Guo and William Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 655
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 657
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Cf. A052709, A052709, A100238.
Cf. A182399, A219534, A000108, A006318.
Cf. A068763.

Table[CatalanNumber[n-1] Hypergeometric2F1[(1-n)/2, -n/2, 3/2-n, -1] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x-4*x^2+x*O(x^n)))/2,n)

a(n) = A052709(n) + A052709(n-1).
A100238(n) = -(-1)^n*a(n), for n>1.
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1)*binomial(n-k, k), where C(q)=binomial(2q, q)/(q+1) are the Catalan numbers (A000108). - Emeric Deutsch, Nov 14 2001 [{a(n+1)}A068763.%20-%20_Wolfdieter%20Lang">{n>=0} = row sum of A068763. - _Wolfdieter Lang, Jan 21 2023]
D-finite with recurrence n*a(n) = (4n-6)*a(n-1)+(4n-12)*a(n-2), n>2. a(1)=1, a(2)=2.
G.f. satisfies A(x)-A(x)^2 = x+x^2. - Ralf Stephan, Jun 30 2003
a(n) = Sum_{k=0..n-1} C(k)*C(k+1, n-k-1). - Paul Barry, Feb 23 2005
G.f. A(x) satisfies A(x)=x+C(2x*A(x)) where C(x) is g.f. of Catalan numbers A000108 offset 1. - Michael Somos, Sep 08 2005
G.f.: (1-sqrt(1-4x-4x^2))/2 = 2(x+x^2)/(1+sqrt(1-4x-4x^2)). - Michael Somos, Jun 08 2000
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([1,2]). - Gary W. Adamson, Oct 27 2008
From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010: (Start)
a(n+1), n >= 0, is column sum for the n-th column of the table R(m,n)=binomial(m+1, n-m)c(m) where c(m) is the m-th Catalan number A000108.
The table entry is nonzero if and only if m <= n <= 2m+1.
R(m,n) gives the number of Rota-Baxter words in one idempotent generator x and one operator of degree m and arity n, or the number of ways of adding m pairs of parentheses to a string of n x's (n necessarily lies between m and 2m+1 inclusive for a nonzero count), such that no two pairs of parentheses are immediately nested and no two x's remain adjacent. (End)
G.f.: A(x) = B(B(x)) where B(x) is the g.f. of A182399. -Paul D. Hanna, Apr 27 2012
G.f.: 1 - x + x*G(0), where G(k) = 1 + 1/(1 - (1+x)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
a(n) ~ (1 + sqrt(2))^(n - 1/2) * 2^(n - 5/4) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 18 2013, simplified Jan 21 2023
O.g.f.: A(x) = x*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318. - Peter Bala, Mar 05 2020
G.f.: A(x) satisfies ((A(x) - A(-x))/(2*x))^2 = S(4*x^2), where S(x) is the g.f. for the large Schröder numbers A006318. - Alexander Burstein, May 20 2021
A(x) = (x + x^2)*c(x+x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Note that (x - x^2)*c(x-x^2) = x. - Peter Bala, Aug 29 2024

A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860

Offset: 0

Philippe Deléham, Jan 19 2004

Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 1,  3,   1;
[4] 0, 1,  6,   6,    1;
[5] 0, 1, 10,  20,   10,    1;
[6] 0, 1, 15,  50,   50,   15,    1;
[7] 0, 1, 21, 105,  175,  105,   21,   1;
[8] 0, 1, 28, 196,  490,  490,  196,  28,  1;
[9] 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;

Indranil Ghosh, Rows 0..100, flattened
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path, The number of double rises of a Dyck path, The number of valleys of a Dyck path, The number of left oriented leafs except the first one of a binary tree, The number of left tunnels of a Dyck path
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Mirror image of triangle A131198. A000108 (row sums, Catalan).
Columns: A000217, A002415, A006542, A006857.
Cf. A001263, A084938, A243752.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n) for x=0,1,2,3,4,5,6,7,8,9. - Philippe Deléham, Aug 10 2006
Sum_{k=0..n} x^(n-k)*T(n,k) = A090192(n+1), A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Oct 21 2006
Sum_{k=0..n} T(n,k)*x^k*(x-1)^(n-k) = A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Oct 20 2007

[[(&+[(-1)^(j-k)*Binomial(2*n-j,j)*Binomial(j,k)*Binomial(2*n-2*j,n-j)/(n-j+1): j in [0..n]]): k in [0..n]]: n in [0..10]];

A090181 := (n,k) -> binomial(n,n-k)*binomial(n-1,n-k)/(n-k+1):
seq(print( seq(A090181(n,k),k=0..n)),n=0..5); # Peter Luschny, May 10 2014
egf := 1+int((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x,x);
s := n -> n!*coeff(series(egf,x,n+2),x,n);
seq(print(seq(coeff(s(n),t,j),j=0..n)),n=0..9); # Peter Luschny, Oct 30 2014
T := proc(n, k) option remember; if k = n or k = 1 then 1 elif k < 1 then 0 else (2*n/k - 1) * T(n-1, k-1) + T(n-1, k) fi end:
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Dec 31 2024

Flatten[Table[Sum[(-1)^(j-k) * Binomial[2n-j,j] * Binomial[j,k] * CatalanNumber[n-j], {j, 0, n}], {n,0,11},{k,0,n}]] (* Indranil Ghosh, Mar 05 2017 *)
p[0, ] := 1; p[1, x] := x; p[n_, x_] := ((2 n - 1) (1 + x) p[n - 1, x] - (n - 2) (x - 1)^2 p[n - 2, x]) / (n + 1);
Table[CoefficientList[p[n, x], x], {n, 0, 9}] // TableForm (* Peter Luschny, Apr 26 2022 *)

c(n) = binomial(2*n,n)/ (n+1);
tabl(nn) = {for(n=0, nn, for(k=0, n, print1(sum(j=0, n, (-1)^(j-k) * binomial(2*n-j,j) * binomial(j,k) * c(n-j)),", ");); print(););};
tabl(11); \\ Indranil Ghosh, Mar 05 2017

from functools import cache
@cache
def Trow(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    if n == 2: return [0, 1, 1]
    A = Trow(n - 2) + [0, 0]
    B = Trow(n - 1) + [1]
    for k in range(n - 1, 1, -1):
        B[k] = (((B[k] + B[k - 1]) * (2 * n - 1)
               - (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 2)) // (n + 1))
    return B
for n in range(10): print(Trow(n)) # Peter Luschny, May 02 2022

def A090181_row(n):
    U = [0]*(n+1)
    for d in DyckWords(n):
        U[d.number_of_peaks()] +=1
    return U
for n in range(8): A090181_row(n) # Peter Luschny, May 10 2014

Triangle T(n, k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938. T(0, 0) = 1, T(n, 0) = 0 for n>0, T(n, k) = C(n-1, k-1)*C(n, k-1)/k for k>0.
Sum_{j>=0} T(n,j)*binomial(j,k) = A060693(n,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*10^k = A143749(n+1). - Philippe Deléham, Oct 14 2008
From Paul Barry, Nov 10 2008: (Start)
Coefficient array of the polynomials P(n,x) = x^n*2F1(-n,-n+1;2;1/x).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j). (End)
Sum_{k=0..n} T(n,k)*5^k*3^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-2)^k = A152681(n); Sum_{k=0..n} T(n,k)*(-1)^k = A105523(n). - Philippe Deléham, Feb 03 2009
Sum_{k=0..n} T(n,k)*2^(n+k) = A156017(n). - Philippe Deléham, Nov 27 2011
T(n, k) = C(n,n-k)*C(n-1,n-k)/(n-k+1). - Peter Luschny, May 10 2014
E.g.f.: 1+Integral((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x dx). - Peter Luschny, Oct 30 2014
G.f.: (1+x-x*y-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x). - Alois P. Heinz, Nov 28 2021, edited by Ron L.J. van den Burg, Dec 19 2021
T(n, k) = [x^k] (((2*n - 1)*(1 + x)*p(n-1, x) - (n - 2)*(x - 1)^2*p(n-2, x))/(n + 1)) with p(0, x) = 1 and p(1, x) = x. - Peter Luschny, Apr 26 2022
Recursion based on rows (see the Python program):
T(n, k) = (((B(k) + B(k-1))*(2*n - 1) - (A(k) - 2*A(k-1) + A(k-2))*(n-2))/(n+1)), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3. # Peter Luschny, May 02 2022

A002002 a(n) = Sum_{k=0..n-1} binomial(n,k+1) * binomial(n+k,k).

Original entry on oeis.org

0, 1, 5, 25, 129, 681, 3653, 19825, 108545, 598417, 3317445, 18474633, 103274625, 579168825, 3256957317, 18359266785, 103706427393, 586889743905, 3326741166725, 18885056428537, 107347191941249, 610916200215241

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Benoit Cloitre, Jan 29 2002: (Start)
Array interpretation (first row and column are the natural numbers):
  1 2 3 ..j ... if b(i,j) = b(i-1,j) + b(i-1,j-1) + b(i,j-1) then a(n+1) = b(n,n)
  2 5 .........
  .............
  i........... b(i,j)
(End)
Number of ordered trees with 2n edges, having root of even degree, nonroot nodes of outdegree at most 2 and branches of odd length. - Emeric Deutsch, Aug 02 2002
Coefficient of x^n in ((1-x)/(1-2x))^n, n>0. - Michael Somos, Sep 24 2003
Number of peaks in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0). Example: a(2)=5 because HH, HU*D, U*DH, UHD, U*DU*D, UU*DD contain 5 peaks (indicated by *). - Emeric Deutsch, Dec 06 2003
a(n) is the total number of HHs in all Schroeder (n+1)-paths. Example: a(2)=5 because UH*HD, H*H*H, UDH*H, H*HUD contain 5 HHs (indicated by *) and the other 18 Schroeder 3-paths contain no HHs. - David Callan, Jul 03 2006
a(n) is the total number of Hs in all Schroeder n-paths. Example: a(2)=5 as the Schroeder 2-paths are HH, DUH, DHU, HDU, DUDU and DDUU, and there are 5 H's. In general, a(n) is the total number of H..Hs (m+1 H's) in all Schroeder (n+m)-paths. - FUNG Cheok Yin, Jun 19 2021
a(n) is the number of points in Z^(n+1) that are L1 (Manhattan) distance <= n from the origin, or the number of points in Z^n that are L1 distance <= n+1 from the origin. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the (n+1)-dimensional cubic lattice as well as the (n+1)-st term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - Shel Kaphan, Dec 25 2022 [Edited by Peter Munn, Jan 05 2023]

			G.f. = x + 5*x^2 + 25*x^3 + 129*x^4 + 681*x^5 + 3653*x^6 + 19825*x^7 + 108545*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 through 100 were computed by Vincenzo Librandi)
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
L. Ericksen, Lattice path combinatorics for multiple product identities, J. of Statistical Planning and Inference 140 (2010) 2113-2226, see p. 2219.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 19.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Bisection of A002003, Cf. A047781, A001003.
a(n)=T(n, n+1), array T as in A050143.
a(n)=T(n, n+1), array T as in A064861.
Half the first differences of central Delannoy numbers (A001850).
a(n)=T(n, n+1), array T as in A008288.
Cf. A026002, A190666, A259554.

[&+[Binomial(n,k+1)*Binomial(n+k,k): k in [0..n]]: n in [0..21]];  // Bruno Berselli, May 19 2011

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n,k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1,k); fi; end; seq(A064861(i,i+1),i=1..40);

CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}],x]  (* Harvey P. Dale, Mar 17 2011 *)
a[ n_] := n Hypergeometric2F1[ n + 1, -n + 1, 2, -1] (* Michael Somos, Aug 09 2011 *)
a[ n_] := With[{m = Abs@n}, Sign[n] Sum[ Binomial[ m, k] Binomial[ m + k - 1, m], {k, m}]]; (* Michael Somos, Aug 09 2011 *)

makelist(sum(binomial(n,k+1)*binomial(n+k,k), k, 0, n), n, 0, 21); /* Bruno Berselli, May 19 2011 */

{a(n) = my(m = abs(n)); sign( n) * sum( k=0, m-1, binomial( m, k+1) * binomial( m+k, k))}; /* Michael Somos, Aug 09 2011 */

/* L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1-x)^(-n)/n! */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)/(1-x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, May 17 2015

a = lambda n: hypergeometric([1-n, -n], [1], 2) if n>0 else 0
[simplify(a(n)) for n in range(22)] # Peter Luschny, Nov 19 2014

G.f.: ((1-x)/sqrt(1-6*x+x^2)-1)/2. - Emeric Deutsch, Aug 02 2002
E.g.f.: exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - Vladeta Jovovic, Mar 28 2004
a(n) = Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k+1). - Paul Barry, Sep 20 2004
a(n) = n * hypergeom([n + 1, -n + 1], [2], -1) = ((n+1)*LegendreP(n+1,3) - (5*n+3)*LegendreP(n,3))/(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
G.f.: x*d/dx log(1/(1-x*A006318(x))). - Vladimir Kruchinin, Apr 19 2011
a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 09 2011
G.f.: -1 + 1 / ( 1 - x / (1 - 4*x / (1 - x^2 / (1 - 4*x / (1 - x^2 / (1 - 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = Sum_{k=0..n} A201701(n,k)^2 = Sum_{k=0..n} A124182(n,k)^2 for n > 0. - Philippe Deléham, Dec 05 2011
D-finite with recurrence: 2*(6*n^2-12*n+5)*a(n-1)-(n-2)*(2*n-1)*a(n-2)-n*(2*n-3)*a(n)=0. - Vaclav Kotesovec, Oct 04 2012
a(n) ~ (3+2*sqrt(2))^n/(2^(5/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
D-finite (an alternative): n*a(n) = (6-n)*a(n-6) + (14*n-72)*a(n-5) + (264-63*n)*a(n-4) + 100*(n-3)*a(n-3) + (114-63*n)*a(n-2) + 2*(7*n-6)*a(n-1), n >= 7. - Fung Lam, Feb 05 2014
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k) and n^3*a(n) = Sum_{k=0..n-1} (4*k^3+4*k^2+4*k+1)*binomial(n-1,k)*binomial(n+k,k). For each of the two equalities, both sides satisfy the same recurrence -- this follows from the Zeilberger algorithm. - Zhi-Wei Sun, Aug 30 2014
a(n) = hypergeom([1-n, -n], [1], 2) for n >= 1. - Peter Luschny, Nov 19 2014
Logarithmic derivative of A001003 (little Schroeder numbers). - Paul D. Hanna, May 17 2015
L.g.f.: L(x) = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1-x)^(-n) / n! = Sum_{n>=1} a(n)*x^n/n where exp(L(x)) = g.f. of A001003. - Paul D. Hanna, May 17 2015
a(n+1) = (1/2^(n+1)) * Sum_{k >= 0} (1/2^k) * binomial(n + k, n)*binomial(n + k, n + 1). - Peter Bala, Mar 02 2017
2*a(n) = A110170(n), n > 0. - R. J. Mathar, Feb 10 2022
a(n) = (LegendreP(n,3) - LegendreP(n-1,3))/2. - Mark van Hoeij, Jul 14 2022
D-finite with recurrence n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Aug 01 2022
From Peter Bala, Nov 08 2022: (Start)
a(n) = (-1)^(n+1)*hypergeom( [n+1, -n+1], [1], 2) for n >= 1.
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 1} binomial(2*n-1, n)*x^n/(1 - x)^(2*n) = x + 5*x^2 + 25*x^3 + 129*x^4 + ....
Row sums of A253283. (End)

More terms from Clark Kimberling

A106228 G.f. satisfies A(x) = 1 + xA(x)/(1 - xA(x)^2).

Original entry on oeis.org

1, 1, 2, 6, 21, 80, 322, 1347, 5798, 25512, 114236, 518848, 2384538, 11068567, 51817118, 244370806, 1159883685, 5536508864, 26560581688, 127993221140, 619280193640, 3007251366000, 14651743202152, 71601107803904, 350873710447210, 1723795243004223

Offset: 0

Paul D. Hanna, May 19 2005

nonn

Number of paths from (0,0) to (3n-3,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have no tripledescents (ddd). Example: a(3)=6 because we have udud, Uddud, udUdd, UddUdd, uudd and Ududd (the remaining four paths contain the string ddd: uUddd, UdUddd, Uuddd and UUdddd; see A027307). - Emeric Deutsch, Jun 08 2005
a(n) = number of node-labeled ordered trees (A000108) on n vertices, each node labeled with a positive integer <= its outdegree. A node is a non-root non-leaf vertex. Example. a(3)=6 counts the 5 ordered trees on 4 vertices with all labels 1 and the tree
.|.
/ \
with its (one and only) node labeled 2. - David Callan, Jul 14 2006
a(n) = number of Schroeder (n-1)-paths with no triple descents. Example: a(4)=21 counts all 22 Schroeder 3-paths (A006318) except UUUDDD. - David Callan, Jul 14 2006
(1 + 2x + 6x^2 + ...)*(1 + x + 2x^2 + 6x^3) = (1 + 3x + 10x^2 + 37x^3 + ...), where A109081 = (1, 1, 3, 10, 37, ...). - Gary W. Adamson, Nov 15 2011
a(n) = number of Motzkin paths of length 2n-1 with no downsteps in odd position. Example: a(3)=6 counts FFFFF, FFUDF, FUFDF, UDFFF, UDUDF, UFFDF with U an upstep (1,1), F a flatstep (1,0), and D a downstep (1,-1). - David Callan, May 20 2015
Number of permutations of length n that avoid 4123, 4132, and 4213. - Jay Pantone, Oct 01 2015
Conjecturally, the number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) and e(i) <= e(k). [Martinez and Savage, 2.21] - Eric M. Schmidt, Jul 17 2017
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>3, 1>4, 4>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is larger than the second element. - Sergey Kitaev, Dec 10 2020
a(n) is the number of peakless Motzkin paths of length 2n that do not start with an up edge and where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on 2n vertices where the leftmost vertex is not matched and only vertices of the same parity can be matched. - Alexander Burstein, May 17 2021

			A = 1 + x*A + x^2*A^3 + x^3*A^5 + x^4*A^7 + x^5*A^9 + ...
a(4) = 21 since the top row terms of Q^3 = (10, 7, 3, 1). - _Gary W. Adamson_, Nov 15 2011
G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1347*x^7 + ...

Jay Pantone, Table of n, a(n) for n = 0..500
Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, and Vincent Vatter, Generating Permutations with Restricted Containers, arXiv:1510.00269 [math.CO], 2015.
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
E. Barcucci, R. Pinzani and R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
N. R. Beaton, M. Bouvel, V. Guerrini, and S. Rinaldi, Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences, Electr. J. Combin. 26 (2019), P3.13. Also at arXiv:1511.04854 [math.CO], 2015-2019.
Beáta Bényi, Toufik Mansour, and José L. Ramírez, Pattern Avoidance in Weak Ascent Sequences, arXiv:2309.06518 [math.CO], 2023.
Wlodzimierz Bryc, Raouf Fakhfakh, and Wojciech Mlotkowski, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality, arXiv:1708.05343 [math.PR], 2017-2019. Also in Probability and Mathematical Statistics (2019), Vol. 39, No. 2, 237-258.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan and Toufik Mansour, Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 62-97.
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, arXiv:2404.04091 [math.CO], 2024. See pp. 2, 22.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

Cf. A027907, A027307, A001006, A086246, A109081, A164965, A262441.

a := proc(n) option remember; if n<2 then 1 elif n=2 then 2 else ((380*n^3-840*n^2+496*n-72)*a(n-1)+(76*n^3-282*n^2+302*n-84)*a(n-2)+(57*n^3-297*n^2+402*n-72)*a(n-3))/(76*n^3-54*n^2-46*n) fi end: seq(a(i),i=0..23); # Peter Luschny, Aug 03 2012

Flatten[{1,Table[1/n*Sum[Binomial[n,k]*Binomial[n+k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 16 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[{-n, 1 - n, n + 1}, {1, 3/2}, 1/4]]; (* Michael Somos, May 27 2014 *)

{a(n)=local(A=1+x+x*O(x^n));for(k=1,n,A=1+x*A/(1-x*A^2)); polcoeff(A,n)}

{a(n) = local(A); if( n<0, 0, n++; A = (1 + x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2; polcoeff( serreverse( x^2 / A), n))}; /* Michael Somos, Jun 18 2005 */

{a(n) = if( n<1, n==0, polcoeff( serreverse( x /  (1 + 2*x + 2*x^2 + x^3) + x * O(x^n)), n))}; /* Michael Somos, Dec 31 2014 */

from mpmath import mp
mp.dps = 32; mp.pretty = True
def A106228(n) : return int(mp.hyper([-n, 1-n, n+1], [1, 3/2], 1/4))
[A106228(n) for n in (0..23)] # Peter Luschny, Aug 02 2012

G.f.: A(x) = (1/x)*series_reversion[x/(1 + x*G001006(x))] and thus G.f. satisfies: A(x) = 1 + x*A(x)*G001006(x*A(x)) where G001006(x) is the g.f. of Motzkin numbers A001006.
G.f.: 1 + x*exp( Sum_{n>=1} A082759(n)*x^n/n ), where A082759(n) = Sum_{k=0..n} binomial(n,k)*trinomial(n,k). - Paul D. Hanna, Nov 02 2012
a(n) = (1/n)sum(binomial(n, j+1)*b(n, j), j=0..n-1), where b(n, j) are the trinomial coefficients [b(n, j)=A027907(n, j)=coefficient of x^j in (1+x+x^2)^n]. - Emeric Deutsch, Jun 08 2005
Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x)) where f(x, y) = y^3 - (1+y)*x*(y-x). - Michael Somos, Jun 18 2005
a(n+1) = Sum[binomial(2n-2k,n-k)*binomial(n+k,n)/(n+1),{k,0,n}]. - David Callan, Aug 16 2006
For n>0: a(n) = 1/n*sum(binomial(n,j)*sum(binomial(j,i)*binomial(n-j,2*j-n-i-1)*2^(2*n-3*j+2*i+1),i=0..n-1), j=0..n); - Vladimir Kruchinin, Dec 26 2010
a(n) = 1/(n+1)*sum(binomial(n+1,k)*binomial(n+k+1,n-k),k,0,n); - Vladimir Kruchinin, Feb 28 2010
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  1, 1, 1
  2, 2, 1, 1
  3, 3, 2, 1, 1
  4, 4, 3, 2, 1, 1
  5, 5, 4, 3, 2, 1, 1
  ...
- Gary W. Adamson, Jul 08 2011
D-finite with recurrence: 4*n*(2*n+1)*a(n) + 2*(6-5*n-10*n^2)*a(n-1) + 12*(-9*n^2+35*n-33)*a(n-2) - 2*(n-3)*(13*n-28)*a(n-3) - 15*(n-3)*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 14 2011
From Gary W. Adamson, Nov 15 2011: (Start)
a(n) is the sum of top row terms of Q^(n-1), where Q = the following infinite square production matrix:
  1, 1, 0, 0, 0, ...
  2, 1, 1, 0, 0, ...
  3, 2, 1, 1, 0, ...
  4, 3, 2, 1, 1, ...
  5, 4, 3, 2, 1, ...
  ... (End)
a(n) = 3_F_2([-n, 1-n, n+1], [1, 3/2], 1/4). - Peter Luschny, Aug 02 2012
A four-term recurrence equation is given in the Maple program. Peter Luschny, Aug 03 2012
a(n) ~ 1/228*sqrt(114)*sqrt((32129+3933*sqrt(57))^(1/3) * ((32129+3933*sqrt(57))^(2/3) + 532 + 38*(32129+3933*sqrt(57))^(1/3))) / ((32129+3933*sqrt(57))^(1/3)) * (((1261+57*sqrt(57))^(2/3) + 112 + 10*(1261+57*sqrt(57))^(1/3)) / (6*(1261+57*sqrt(57))^(1/3)))^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 16 2013
G.f. satisfies x*F(x)^3 - x*F(x)^2 + (x-1)*F(x) + 1 = 0. - Jay Pantone, Oct 01 2015
G.f. satisfies A(-x*A(x)^3) = 1/A(x). - Alexander Burstein, Dec 05 2019
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A262441. - Seiichi Manyama, Mar 31 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n/2+3*k/2+1/2,n)/(n+3*k+1). - Seiichi Manyama, Apr 04 2024

A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 5, 1, 10, 30, 35, 14, 1, 15, 70, 140, 126, 42, 1, 21, 140, 420, 630, 462, 132, 1, 28, 252, 1050, 2310, 2772, 1716, 429, 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430, 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862

Offset: 0

N. J. A. Sloane, Nov 23 2003

Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.
T(n,k) is number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g., T(2,1)=3 because we have UHD, UDH and HUD. - Emeric Deutsch, Dec 06 2003
Little Schroeder numbers A001003 have a(n) = Sum_{k=0..n} A088617(n,k)*(-1)^(n-k)*2^k. - Paul Barry, May 24 2005
Conjecture: The expected number of U's in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - David Callan, Jul 25 2008
T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
The antidiagonals of this lower triangular matrix are the rows of A055151. - Tom Copeland, Jun 17 2015

			Triangle begins:
  [0] 1;
  [1] 1,  1;
  [2] 1,  3,   2;
  [3] 1,  6,  10,    5;
  [4] 1, 10,  30,   35,    14;
  [5] 1, 15,  70,  140,   126,    42;
  [6] 1, 21, 140,  420,   630,   462,   132;
  [7] 1, 28, 252, 1050,  2310,  2772,  1716,   429;
  [8] 1, 36, 420, 2310,  6930, 12012, 12012,  6435,  1430;
  [9] 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only]
A. Kirillov, On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials, arXiv preprint arXiv:1502.00426 [math.RT], 2016.
M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.
Paul W. Lapey and Aaron Williams, A Shift Gray Code for Fixed-Content Łukasiewicz Words, Williams College, 2022.
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.
Jason P. Smith, The poset of graphs ordered by induced containment, arXiv:1806.01821 [math.CO], 2018.

Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, A001700, A002457, A002802, A002803.
Cf. A000108, A001003, A006318, A060693, A084938, A085478.
Cf. A033282, A055151, A123160.
Cf. A001263, A011973, A055151, A060693, A074909, A086810, A107131.

[[Binomial(n+k,n)*Binomial(n,k)/(k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 18 2015

R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n,0,10}, {k,0,n}]//Flatten (* Michael De Vlieger, Aug 10 2017 *)

{T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}

flatten([[binomial(n+k, 2*k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 22 2022

Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - Philippe Deléham, Dec 05 2003
Sum_{k=0..n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Aug 18 2005
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Oct 18 2007
O.g.f. (with initial 1 excluded) is the series reversion with respect to x of (1-t*x)*x/(1+x). Cf. A062991 and A089434. - Peter Bala, Jul 31 2012
G.f.: 1 + (1 - x - T(0))/y, where T(k) = 1 - x*(1+y)/( 1 - x*y/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
From Peter Bala, Jul 20 2015: (Start)
O.g.f. A(x,t) = ( 1 - x - sqrt((1 - x)^2 - 4*x*t) )/(2*x*t) = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2 + ....
1 + x*(dA(x,t)/dx)/A(x,t) = 1 + (1 + t)*x + (1 + 4*t + 3*t^2)*x^2 + ... is the o.g.f. for A123160.
For n >= 1, the n-th row polynomial equals (1 + t)/(n+1)*Jacobi_P(n-1,1,1,2*t+1). Removing a factor of 1 + t from the row polynomials gives the row polynomials of A033282. (End)
From Tom Copeland, Jan 22 2016: (Start)
The o.g.f. G(x,t) = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/2x = (t + t^2) x + (t + 3t^2 + 2t^3) x^2 + (t + 6t^2 + 10t^3 + 5t^3) x^3 + ... generating shifted rows of this entry, excluding the first, was given in my 2008 formulas for A033282 with an o.g.f. f1(x,t) = G(x,t)/(1+t) for A033282. Simple transformations presented there of f1(x,t) are related to A060693 and A001263, the Narayana numbers. See also A086810.
The inverse of G(x,t) is essentially given in A033282 by x1, the inverse of f1(x,t): Ginv(x,t) = x [1/(t+x) - 1/(1+t+x)] = [((1+t) - t) / (t(1+t))] x - [((1+t)^2 - t^2) / (t(1+t))^2] x^2 + [((1+t)^3 - t^3) / (t(1+t))^3] x^3 - ... . The coefficients in t of Ginv(xt,t) are the o.g.f.s of the diagonals of the Pascal triangle A007318 with signed rows and an extra initial column of ones. The numerators give the row o.g.f.s of signed A074909.
Rows of A088617 are shifted columns of A107131, whose reversed rows are the Motzkin polynomials of A055151, related to A011973. The diagonals of A055151 give the rows of A088671, and the antidiagonals (top to bottom) of A088617 give the rows of A107131 and reversed rows of A055151. The diagonals of A107131 give the columns of A055151. The antidiagonals of A088617 (bottom to top) give the rows of A055151.
(End)
T(n, k) = [x^k] hypergeom([-n, 1 + n], [2], -x). - Peter Luschny, Apr 26 2022

A000354 Expansion of e.g.f. exp(-x)/(1-2*x).

Original entry on oeis.org

1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561, 112690097, 2253801941, 49583642701, 1190007424825, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 28276861228096781665, 1017967004211484139941, 38682746160036397317757

Offset: 0

N. J. A. Sloane

a(n) is the permanent of the n X n matrix with 1's on the diagonal and 2's elsewhere. - Yuval Dekel, Nov 01 2003. Compare A157142.
Starting with offset 1 = lim_{k->infinity} M^k, where M = a tridiagonal matrix with (1,0,0,0,...) in the main diagonal, (1,3,5,7,...) in the subdiagonal and (2,4,6,8,...) in the subsubdiagonal. - Gary W. Adamson, Jan 13 2009
a(n) is also the number of (n-1)-dimensional facet derangements for the n-dimensional hypercube. - Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009
a(n) is the number of ways to write down each n-permutation and underline some (possibly none or all) of the elements that are not fixed points. a(n) = Sum_{k=0..n} A008290(n,k)*2^(n-k). - Geoffrey Critzer, Dec 15 2012
Type B derangement numbers: the number of fixed point free permutations in the n-th hyperoctahedral group of signed permutations of {1,2,...,n}. See Chow 2006. See A000166 for type A derangement numbers. - Peter Bala, Jan 30 2015

			G.f. = 1 + x + 5*x^2 + 29*x^3 + 233*x^4 + 2329*x^5 + 27949*x^6 + 391285*x^7 + ... - _Michael Somos_, Apr 14 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Chak-On Chow, On derangement polynomials of type B, Séminaire Lotharingien de Combinatoire 55 (2006), Article B55b.
Gary Gordon and Elizabeth McMahon, Moving faces to other places: Facet derangements, arXiv:0906.4253 [math.CO], 2009.
Gary Gordon and Elizabeth McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
Édouard Lucas, Théorie des nombres (annotated scans of a few selected pages)
István Mezo, Victor H. Moll, José L. Ramírez, and Diego Villamizar, On the r-Derangements of type B, arXiv:2103.04151 [math.CO], 2021.
István Mező, Victor H. Moll, José Ramírez, and Diego Villamizar, On the r-derangements of type B, Online Journal of Analytic Combinatorics, Issue 16 (2021), #05.
Jean-Christophe Pain, A sum rule for r-derangements obtained from the Cauchy product of exponential generating functions, arXiv:2408.15927 [math.CO], 2024.
Simon Plouffe, Exact formulas for integer sequences
L. W. Shapiro & N. J. A. Sloane, Correspondence, 1976
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A061714, A008290, A000166, A113012, A263895.

Column k=2 of A320032.

a := n -> (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)):
seq(simplify(a(n)), n=0..18); # Peter Luschny, May 09 2017
a := n -> 2^n*add((n!/k!)*(-1/2)^k, k=0..n):
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2020
seq(simplify(2^n*KummerU(-n, -n, -1/2)), n = 0..19); # Peter Luschny, May 10 2022

FunctionExpand @ Table[ Gamma[ n+1, -1/2 ]*2^n/Exp[ 1/2 ], {n, 0, 24}]
With[{nn=20},CoefficientList[Series[Exp[-x]/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 22 2013 *)
a[n_] := 2^n n! Sum[(-1)^i/(2^i i!), {i, 0, n}]; Table[a[n], {n, 0, 20}] (* Gerry Martens, May 06 2016 *)
a[ n_] := If[ n < 1, Boole[n == 0], (2 n - 1) a[n - 1] + (2 n - 2) a[n - 2]]; (* Michael Somos, Sep 28 2017 *)
a[ n_] := Sum[ (-1)^(n + k) Binomial[n, k] k! 2^k, {k, 0, n}]; (* Michael Somos, Apr 14 2018 *)
a[ n_] := If[ n < 0, 0, (2^n Gamma[n + 1, -1/2]) / Sqrt[E] // FunctionExpand]; (* Michael Somos, Apr 14 2018 *)
a[n_] := n! 2^n Hypergeometric1F1[-n, -n, -1/2];
Table[a[n], {n, 0, 19}]   (* Peter Luschny, Jul 28 2024 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-2*x))) \\ Joerg Arndt, Apr 15 2013

vector(100, n, n--; sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k)) \\ Altug Alkan, Oct 30 2015

{a(n) = if( n<1, n==0, (2*n - 1) * a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Sep 28 2017 */

Inverse binomial transform of double factorials A000165. - Paul Barry, May 26 2003
a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*k!*2^k. - Paul Barry, May 26 2003
a(n) = Sum_{k=0..n} A008290(n, k)*2^(n-k). - Philippe Deléham, Dec 13 2003
a(n) = 2*n*a(n-1) + (-1)^n, n > 0, a(0)=1. - Paul Barry, Aug 26 2004
D-finite with recurrence a(n) = (2*n-1)*a(n-1) + (2*n-2)*a(n-2). - Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009
From Groux Roland, Jan 17 2011: (Start)
a(n) = (1/(2*sqrt(exp(1))))*Integral_{x>=-1} exp(-x/2)*x^n dx;
Sum_{k>=0} 1/(k!*2^(k+1)*(n+k+1)) = (-1)^n*(a(n)*sqrt(exp(1))-2^n*n!). (End)
a(n) = round(2^n*n!/exp(1/2)), x >= 0. - Simon Plouffe, Mar 1993
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
From Peter Bala, Jan 30 2015: (Start)
a(n) = Integral_{x = 0..inf} (2*x - 1)^n*exp(-x) dx.
b(n) := 2^n*n! satisfies the recurrence b(n) = (2*n - 1)*b(n-1) + (2*n - 2)*b(n-2), the same recurrence as satisfied by a(n). This leads to the continued fraction representation a(n) = 2^n*n!*( 1/(1 + 1/(1 + 2/(3 + 4/(5 +...+ (2*n - 2)/(2*n - 1) ))))) for n >= 2, which in the limit gives the continued fraction representation sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). (End)
For n > 0, a(n) = 1 + 4*Sum_{k=0..n-1} A263895(n). - Vladimir Reshetnikov, Oct 30 2015
a(n) = (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)). - Peter Luschny, May 09 2017
a(n+1) >= A113012(n). - Michael Somos, Sep 28 2017
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(n) = 2^n*KummerU(-n, -n, -1/2). - Peter Luschny, May 10 2022
a(n) = 2^n*n!*hypergeom([-n], [-n], -1/2). - Peter Luschny, Jul 28 2024

cp's OEIS Frontend

A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.

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A027307 Number of paths from (0,0) to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

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A027941 a(n) = Fibonacci(2*n + 1) - 1.

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A025227 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 3.

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A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

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A025227 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 3.

A106228 G.f. satisfies A(x) = 1 + xA(x)/(1 - xA(x)^2).