A001850 -id:A001850 - cp's OEIS frontend

A036442 a(n) = 2^((n-1)*(n+2)/2).

1, 4, 32, 512, 16384, 1048576, 134217728, 34359738368, 17592186044416, 18014398509481984, 36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, 20282409603651670423947251286016, 664613997892457936451903530140172288

Offset: 1

Abdallah Rayhan (rayhan(AT)engr.uvic.ca)

Number of redundant paths for a fault-tolerant ATM switch.
Hankel transform (see A001906 for definition ) of A001850, A006139, A084601; also Hankel transform of the sequence 1, 0, 4, 0, 24, 0, 160, 0, 1120, ... (A059304 with interpolated zeros). - Philippe Deléham, Jul 03 2005
Hankel transform of A109980. Unsigned version of A127945. - Philippe Deléham, Dec 11 2008
a(n) = the multiplicative Wiener index of the wheel graph with n+3 vertices. The multiplicative Wiener index of a connected simple graph G is defined as the product of the distances between all pairs of distinct vertices of G. The wheel graph with n+3 vertices has (n+3)(n+2)/2 pairs of distinct vertices, of which 2(n+2) are adjacent; each of the remaining (n+2)(n-1)/2 pairs are at distance 2; consequently, the multiplicative Wiener index is 2^((n-1)(n+2)/2) = a(n). - Emeric Deutsch, Aug 17 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..80
P. Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161, 2013 and J. Int. Seq. 17 (2014) # 14.5.1
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
C. Lo and C. Chiu, A Fault-Tolerant Architecture for ATM Networks, 20th IEEE Conf. Local Computer Networks, 1995, pp. 29-36
Index to divisibility sequences

I:=[1]; [n le 1 select I[n] else Self(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Oct 24 2012

Table[2^((n-1) * (n+2)/2), {n, 1, 30}] (* Vincenzo Librandi, Oct 24 2012 *)

A036442[n]:=2^((n-1)*(n+2)/2)$
makelist(A036442[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

a(n)=2^((n-1)*(n+2)/2) \\ Charles R Greathouse IV, Oct 24 2012

a(1) = 1, a(n) = a(n-1) * 2^n. - Vincenzo Librandi, Oct 24 2012

A049600 Array T read by diagonals; T(i,j) is the number of paths from (0,0) to (i,j) consisting of nonvertical segments (x(k),y(k))-to-(x(k+1),y(k+1)) such that 0 = x(1) < x(2) < ... < x(n-1) < x(n)=i, 0 = y(1) <= y(2) <= ... <= y(n-1) <= y(n)=j, for i >= 0, j >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 4, 8, 8, 0, 1, 5, 13, 20, 16, 0, 1, 6, 19, 38, 48, 32, 0, 1, 7, 26, 63, 104, 112, 64, 0, 1, 8, 34, 96, 192, 272, 256, 128, 0, 1, 9, 43, 138, 321, 552, 688, 576, 256, 0, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512

Offset: 0

Clark Kimberling

Essentially array A059576 divided by sequence A011782.
[Hetyei] calls a variant of this array (omitting the initial row of zeros) the asymmetric Delannoy numbers and shows how they arise in certain lattice path enumeration problems and a face enumeration problem associated to Jacobi polynomials. - Peter Bala, Oct 29 2008
Essentially triangle in A208341. - Philippe Deléham, Mar 23 2012
T(n+k,n) is the dot product of a vector from the n-th row of Pascal's triangle A007318 with a vector created by the first n+1 values evaluated from the cycle index of symmetry group S(k). Example: T(4+3,4) = T(7,4) = {1,4,6,4,1}.{1,4,10,20,35} = 192. - Richard Turk, Sep 21 2017
The formula T(n,k) = Sum_{r=0..n-1} C(k+r,r)*C(n-1,r) (Paul D. Hanna, Oct 06 2006) counts the paths of the title by number, r, of interior vertices in the path. - David Callan, Nov 25 2021

			Diagonals (each starting on row 1): {0}; {0,1}; {0,1,2}; ...
Array begins:
    0     0     0     0     0     0     0     0     0     0     0 ...
    1     1     1     1     1     1     1     1     1     1     1 ...
    2     3     4     5     6     7     8     9    10    11    12 ...
    4     8    13    19    26    34    43    53    64    76    89 ...
    8    20    38    63    96   138   190   253   328   416   518 ...
   16    48   104   192   321   501   743  1059  1462  1966  2586 ...
   32   112   272   552  1002  1683  2668  4043  5908  8378 11584 ...
   64   256   688  1520  2972  5336  8989 14407 22180 33028 47818 ...
Triangle begins:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3,  4;
  0, 1, 4,  8,  8;
  0, 1, 5, 13, 20,  16;
  0, 1, 6, 19, 38,  48,  32;
  0, 1, 7, 26, 63, 104, 112, 64;
  ...
(1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) where DELTA is the operator defined in A084938 begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  4,  0;
  1, 4,  8,  8,   0;
  1, 5, 13, 20,  16,   0;
  1, 6, 19, 38,  48,  32,  0;
  1, 7, 26, 63, 104, 112, 64, 0;

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
David Callan, Some bijections for lattice paths, arXiv:2112.05241 [math.CO], 2021.
David Callan, A bijection for Delannoy paths, arXiv:2202.04649 [math.CO], 2022.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Robert Cori and Gabor Hetyei, Genus one partitions, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, pp. 333-344, .
Sergio Falcon, On the complex k-Fibonacci numbers, Cogent Mathematics, (2016), 3: 1201944. See Table 1.
G. Hetyei, Central Delannoy numbers, Legendre polynomials and a balanced join operation preserving the Cohen-Macauley property, Annals of Combinatorics, 10 (2006), 443-462.
G. Hetyei, Central Delannoy numbers and balanced Cohen-Macaulay complexes, Ann. Comb. 10 (2006), 443-462.
G. Hetyei, Links we almost missed between Delannoy numbers and Legendre polynomials
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Clark Kimberling, Enumeration of paths, compositions of integers and Fibonacci numbers, Fib. Quarterly 39 (5) (2001) 430-435, Figure 1.
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3C.
Benjamin Lyons and McCabe Olsen, Self-Reachable Chip Configurations on Trees, arXiv:2409.00763 [math.CO], 2024. See p. 18.
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Diagonal sums are even-indexed Fibonacci numbers. Alternating (+-) diagonal sums are signed Fibonacci numbers.
T(n, n-1) = A001850(n) (Delannoy numbers). T(n, n)=A047781. Cf. A035028, A055587.
Cf. A208341. A055587.

a049600 n k = a049600_tabl !! n !! k
a049600_row n = a049600_tabl !! n
a049600_tabl = [0] : map (0 :) a208341_tabl
-- Reinhard Zumkeller, Apr 15 2014

A049600 := proc(n,k)
    add(binomial(k+j,j)*binomial(n-1,j),j=0..n-1) ;
end proc: # R. J. Mathar, Oct 26 2015

t[n_, k_] := Hypergeometric2F1[ n-k+1, 1-k, 1, -1] // Floor; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
t[n_, k_] := Sum[LaguerreL[n-k, i, 0]* LaguerreL[k-i, i, 0], {i,0,k}] //Floor; Table[t[n,k], {n, 0, 16}, {k, -1, n}] (* Richard Turk, Sep 08 2017 *)
T[n_, k_] := If[k == 0, 0, JacobiP[k - 1, 0, 1 - 2*k + n, 3]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 25 2021 *)

{A(i, j) = polcoeff( (x / (1 - 2*x)) * ((1 - x) / (1 - 2*x))^j + x * O(x^i), i)}; /* Michael Somos, Oct 01 2003 */

T(n,k)=sum(j=0,n-1,binomial(k+j,j)*binomial(n-1,j)) \\ Paul D. Hanna, Oct 06 2006

T(n,k) = Sum_{j=0..n-1} C(k+j,j)*C(n-1,j). - Paul D. Hanna, Oct 06 2006
T(i,j) = 2*T(i-1,j) + T(i,j-1) - T(i-1,j-1) with T(0,0)=1 and T(i,j)=0 if one of i,j<0. - Theodore Kolokolnikov, Jul 05 2010
O.g.f.: t*x/(1 - (2*t+1)*x + t*x^2) = t*x + (t + 2*t^2)*x^2 + (t + 3*t^2 + 4*t^3)*x^3 + .... Taking the row reverse of this triangle (with an additional column of 1's) gives A055587. - Peter Bala, Sep 10 2012
T(i,0) = 2^(i-1) and for j>0, T(i,j) = T(i,j-1) + Sum_{k=0..i-1} T(k,j). - Glen Whitney, Aug 17 2021
T(n, k) = JacobiP(k - 1, 0, 1 - 2*k + n, 3) for k >= 1. - Peter Luschny, Nov 25 2021

A120588 G.f. is 1 + x*c(x), where c(x) is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152

Offset: 0

Paul D. Hanna, Jun 16 2006, Jan 24 2008

Previous name was: G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, with A(0) = 1.
This is essentially a duplicate of entry A000108, the Catalan numbers (a(n) = A000108(n-1) for n>0).
In order for the g.f. of an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n, where n > 1, it is necessary that the sequence start with [1, d, m*n*(n-1)/2], where d divides m*n*(n-1)/2 (m>0) and that the coefficients are given by r = n + d^2/m, c = r-1 and b = d^3/m. The remaining terms may then be integer and still satisfy: a_n(k) = r*a(k), where a_n(k) is the k-th term of the n-th self-convolution of the sequence.

			A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 +...
A(x)^3 = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 42*x^5 + 126*x^6 + 396*x^7 +..
More generally, given the functional equation:
r*A(x) = r-1 + b*x + A(x)^n
the series solution is:
A(x) = Sum_{i>=0} C(n*i,i)/(n*i-i+1)*(r-1+bx)^(n*i-i+1)/r^(n*i+1)
which can be expressed as:
A(x) = G( (r-1+bx)^(n-1)/r^n ) * (r-1+bx)/r
where G(x) satisfies: G(x) = 1 + x*G(x)^n .
Also we have:
A(x) = 1 + Series_Reversion[ (1 + r*x - (1+x)^n )/b ].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A120589 (A(x)^2); A120590 - A120607.

Cf. A001850, A026641.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (3 - Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 18 2019

a[ n_] := SeriesCoefficient[ 1 + (1 - Sqrt[1 - 4 x]) / 2, {x, 0, n}]; (* Michael Somos, May 18 2015 *)

{a(n)=local(A=1+x+x^2+x*O(x^n));for(i=0,n,A=A-3*A+2+x+A^2);polcoeff(A,n)}

{a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

((3-sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

G.f.: A(x) = 1 + Series_Reversion(1+3*x - (1+x)^2).
Lagrange Inversion yields g.f.: A(x) = Sum_{n>=0} C(2*n,n)/(n+1)*(2+x)^(n+1)/3^(2*n+1).
G.f.: (3 - sqrt(1-4*x))/2. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
a(n) = Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
G.f.: 2 - G(0), where G(k)= 2*x*(2*k+1) + k +1 - 2*x*(k+1)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013
G.f.: 2 - G(0), where G(k)= 1 - x/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jul 19 2013
a(n) ~ 2^(2*n-2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 19 2013
Given g.f. A(x), A001850(n-1) = coefficient of x^n in A(x)^n if n>0, the derivative of log(A(x)) is the g.f. for A026641. - Michael Somos, May 18 2015
A(x) = (1 + 2*Sum_{n >= 1} Catalan(n)*x^n)/(1 + Sum_{n >= 1} Catalan(n)*x^n) = (1 + 3/2*Sum_{n >= 1} binomial(2*n,n)*x^n )/(1 + Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016
D-finite with recurrence n*a(n) +2*(-2*n+3)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

New name by Wolfdieter Lang, Feb 06 2020

A069835 Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).

Original entry on oeis.org

1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544, 98365972, 700701808, 5011371964, 35961808432, 258805997752, 1867175631136, 13500088649734, 97794850668952, 709626281415076, 5157024231645616, 37528209137458516, 273431636191026064, 1994448720786816712

Offset: 0

Benoit Cloitre, May 03 2002

2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 - 4*k*x + 4*x^2) and e.g.f. exp(2*k*x)BesselI(0, 2*sqrt(k^2-1)*x). - Paul Barry, May 25 2005
Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - Paul Barry, May 25 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have three colors and H steps can have four colors. - N-E. Fahssi, Mar 31 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and two kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Hankel transform is 2^n*3^C(n+1,2) = (-1)^C(n+1,2)*A127946(n). - Paul Barry, Jan 24 2011
Central terms of triangle A152842. - Reinhard Zumkeller, May 01 2014
Diagonal of rational functions 1/(1 - x - y - 2*x*y), 1/(1 - x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 06 2018
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

			The array b is a rewriting of A081577:
  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  1,  4,  7,  10,  13,   16,   19,   22,    25,    28,    31, ...
  1,  7, 22,  46,  79,  121,  172,  232,   301,   379,   466, ...
  1, 10, 46, 136, 307,  586, 1000, 1576,  2341,  3322,  4546, ...
  1, 13, 79, 307, 886, 2086, 4258, 7834, 13327, 21331, 32521, ...

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..250
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Cf. A001850.

List([0..25],n->Sum([0..n],k->Binomial(n,k)^2*3^k)); # Muniru A Asiru, Jul 29 2018

a069835 n = a081577 (2 * n) n -- Reinhard Zumkeller, Mar 16 2014

Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

PARI
```
a(n)=sum(k=0,n,binomial(n,k)^2*3^k)
```

a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [0,1], [1,1], [1,1]]; /* note the double [1,1] */
\\ Joerg Arndt, Jul 01 2011

a(n)=pollegendre(n,2)<Charles R Greathouse IV, Mar 18 2017

From Vladeta Jovovic, May 13 2003: (Start)
a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2.
D-finite with recurrence: a(n) = 4*(2*n-1)/n*a(n-1) - 4*(n-1)/n*a(n-2).
G.f.: 1/sqrt(1 - 8*x + 4*x^2). (End)
a(n) equals the central coefficient of (1 + 4*x + 3*x^2)^n. - Paul D. Hanna, Jun 03 2003
E.g.f.: exp(4*x)*Bessel_I(0, 2*sqrt(3)*x). - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2*(n-k), n)*(-1)^k*2^(n-2*k). - Paul Barry, May 25 2005
a(n) = Sum_{k=0..n} C(n, k)*C(n+k, k)*2^(n-k). - Paul Barry, May 25 2005
a(n) = Sum_{k=0..n} C(n, k)^2*3^k. - Paul Barry, Oct 15 2005
G.f.: 1/(1-4x-6x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-... (continued fraction). - Paul Barry Jan 24 2011
Asymptotic: a(n) ~ sqrt(1/2 + 1/sqrt(3))*(1+sqrt(3))^(2*n)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 11 2012
0 = a(n)*(16*a(n+1) - 48*a(n+2) + 8*a(n+3)) + a(n+1)*(-16*a(n+1) + 64*a(n+2) - 12*a(n+3)) + a(n+2)*(-4*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Apr 21 2020

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, 1, 8, 30, 68, 90, 1, 10, 48, 146, 304, 394, 1, 12, 70, 264, 714, 1412, 1806, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098

Offset: 1

N. J. A. Sloane

A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - Philippe Deléham, May 15 2005
From Johannes W. Meijer, Sep 22 2010, Jul 15 2013: (Start)
Note that for the terms T(n,k) of this triangle n indicates the column and k the row.
The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.
Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) =  A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)
T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - Jacob Post, Jun 19 2018

			Triangle starts:
  1;
  1,    2;
  1,    4,    6;
  1,    6,   16,   22;
  1,    8,   30,   68,   90;
  1,   10,   48,  146,  304,  394;
  1,   12,   70,  264,  714, 1412, 1806;
  ... - _Joerg Arndt_, Sep 29 2013

T. D. Noe, Rows k = 1..50 of triangle, flattened
Henry Bottomley, Illustration of initial terms
Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
J. M. Oh, An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89, last table.
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.
Cf. A084938, A103885, A238112, A330801.
Cf. A001003 (row sums), A026003 (antidiagonal sums).
Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).

a033877 n k = a033877_tabl !! n !! k
a033877_row n = a033877_tabl !! n
a033877_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Apr 17 2013

function t(n,k)
  if k le 0 or k gt n then return 0;
  elif k eq 1 then return 1;
  else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
  end if;
end function;
[t(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 23 2023

T := proc(n, k) option remember; if n=1 then return(1) fi; if kJohannes W. Meijer, Sep 22 2010, revised Jul 17 2013

Mathematica
```
T[1, ]:= 1; T[n, k_]/;(k
				
```

def A033877_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A033877_row(n)) # Peter Luschny, Mar 16 2016

@CachedFunction
def t(n, k): # t = A033847
    if (k<0 or k>n): return 0
    elif (k==1): return 1
    else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 23 2023

As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.
G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic, Feb 16 2003
Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - Johannes W. Meijer, Sep 28 2013
From G. C. Greubel, Mar 23 2023: (Start)
(t(n, k) as a lower triangle)
t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.
t(n, n) = A006318(n-1).
t(2*n-1, n) = A330801(n-1).
t(2*n-2, n) = A103885(n-1), n > 1.
Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.
Sum_{k=1..n} t(n, k) = A001003(n).
Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.
Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).
Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)

More terms from David W. Wilson

A002003 a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).

Original entry on oeis.org

0, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of order-preserving partial self maps of {1,...,n}. For example, a(2) = 8 because there are 8 order-preserving partial self maps of {1,2}: (1 2), (1 1), (2 2), (1 -), (2 -), (- 1), (- 2), (- -). Here for example (2 -) represents the partial map which maps 1 to 2 but does not include 2 in its domain. - James East, Oct 25 2005
From Peter Bala, Mar 02 2020: (Start)
For fixed m = 1,2,3,..., we conjecture that the sequence b(n) := a(m*n) satisfies a recurrence of the form P(2*m,n)*b(n+1) + P(2*m,-n)*b(n-1) = Q(2*m,n)*b(n), where the polynomials P(2*m,n) and Q(2*m,n) have degree 2*m. Conjecturally, the polynomial Q(2*m,n) is an even function of n; its 2*m zeros seem to belong to the interval [-1, 1] and 2*m - 2 of these zeros appear to lie close to the rational numbers of the form +-(2*k + 1)/(2*m), where 0 <= k <= m - 2. Cf. A103885. (End)
a(n), n>0, is the number of points at L1 distance = n from any given point in Z^n.  The sequence is also the difference between the central diagonal (A001850) and +-1 diagonal (A002002) of the Delannoy number triangle (A008288).  - Shel Kaphan, Feb 15 2023

			G.f. = 2*x + 8*x^2 + 38*x^3 + 192*x^4 + 1002*x^5 + 5336*x^6 + 28814*x^7 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Peter Bala, A supercongruence for A002003
J. Brzozowski, M. Szykula, Large Aperiodic Semigroups, arXiv preprint arXiv:1401.0157 [cs.FL], 2013-2014.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Cf. A002002, A006318, A027307, A108424, A103885.

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);

Flatten[{0,Table[SeriesCoefficient[((1+x)/Sqrt[1-6*x+x^2]-1)/2,{x,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 04 2012 *)
a[ n_] := If[ n < 1, 0, Hypergeometric2F1[ n, -n, 1, -1]]; (* Michael Somos, Aug 24 2014 *)
Table[2*Sum[Binomial[n-1,k]Binomial[n+k,k],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Sep 18 2024 *)

{a(n) = if( n<1, 0, polcoeff( ((1 - x^2) / (1 - x)^2 + x * O(x^n))^n, n))} /* Michael Somos, Sep 24 2003 */

from math import comb
def A002003(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

a(n) = 2*A047781(n).
From Vladeta Jovovic, Mar 28 2004: (Start)
G.f.: ((1+x)/sqrt(1-6*x+x^2)-1)/2.
E.g.f.: exp(3*x)*(2*BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). (End)
a(n) = T(n, n-1), array T as in A064861.
a(n) = T(n, n-2), array T as in A049600.
a(n+1) = A110110(2n+1). - Tilman Neumann, Feb 05 2009
a(n) = 2 * JacobiP(n-1,0,1,3) = ((7*n+3)*LegendreP(n,3) - (n+1)*LegendreP(n+1,3)) /(2*n) for n > 0. - Mark van Hoeij, Jul 12 2010
Logarithmic derivative of A006318, the large Schroeder numbers. - Paul D. Hanna, Oct 25 2010
D-finite with recurrence: 4*(3*n^2-6*n+2)*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^(3/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012
Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n>=7. - Fung Lam, Feb 05 2014
a(n) = Hyper2F1([-n, n], [1], -1) for n > 0. - Peter Luschny, Aug 02 2014
a(n) = [x^n] ((1+x)/(1-x))^n for n > 0. - Seiichi Manyama, Jun 07 2018
From Peter Bala, Mar 13 2020: (Start)
a(n) = 2 * Sum_{k = 0..n-1}  2^k*C(n,k+1)*C(n-1,k).
a(n) = 2 * (-1)^(n+1) * Sum_{k = 0..n-1} (-2)^k*C(n+k,n-1)*C(n-1,k).
a(n) = Sum_{k = 0..n} C(n,k)*C(2*n-k-1,n-1).
Conjecture: a(n) = - [x^n] (1 - F(x))^n, where F(x) = 2*x + 6*x^2 + 34*x^3 + 238*x^4 + ... is the o.g.f. of A108424. Equivalently, a(n) = -[x^n](G(x))^(-n), where G(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 498*x^4 + ... is the o.g.f. of A027307.
a(p) == 2 ( mod p^3 ) for prime p >= 5. (End)
a(n) = Sum_{k = 1..n} C(n, k) * C(n-1, k-1) * 2^k. - Michael Somos, May 23 2021
a(n) = A001850(n) - A002002(n), for n > 0. - Shel Kaphan, Feb 15 2023

More terms from Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10 2001

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

Original entry on oeis.org

1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725

Offset: 0

Paul D. Hanna, Jun 10 2003

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - N-E. Fahssi, Mar 30 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Sums of squares of coefficients of (1+2*x)^n. - Joerg Arndt, Jul 06 2011
The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007
Partial sums of A085363. - J. M. Bergot, Jun 12 2013
Diagonal of rational functions 1/(1 - x - y - 3*x*y), 1/(1 - x - y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Tewodros Amdeberhan, In search of multiple expressions for a sequence
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Curtis Greene, Posets of shuffles, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).
Christopher Huffaker, Nathan McCue, Cameron N. Miller, and Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], 2015.
Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See p. 3.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Cf. A001850, A059231, A059304, A246923 (a(n)^2).

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*4^k)); # Muniru A Asiru, Jul 29 2018

[3^n*Evaluate(LegendrePolynomial(n), 5/3) : n in [0..40]]; // G. C. Greubel, May 30 2023

seq(simplify(hypergeom([-n,1/2], [1], -8)),n=0..19); # Peter Luschny, Apr 26 2016

Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *)
Table[Hypergeometric2F1[-n, -n, 1, 4], {n,0,30}] (* Vladimir Reshetnikov, Nov 29 2013 *)
CoefficientList[Series[1/Sqrt[1-10x+9x^2],{x,0,30}],x] (* Harvey P. Dale, Mar 08 2016 *)
Table[3^n*LegendreP[n, 5/3], {n, 0, 40}] (* G. C. Greubel, May 30 2023 *)

{a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0,n, binomial(n, k)^2 * 4^k))}; /* Michael Somos, Oct 08 2003 */

{a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* Michael Somos, Oct 08 2003 */

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [0,1], [1,1], [1,1], [1,1]]; /* note the triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

a(n)={local(v=Vec((1+2*x)^n));sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2+imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */

[3^n*gen_legendre_P(n, 0, 5/3) for n in range(41)] # G. C. Greubel, May 30 2023

G.f.: 1 / sqrt(1 - 10*x + 9*x^2).
From Vladeta Jovovic, Aug 20 2003: (Start)
Binomial transform of A059304.
G.f.: Sum_{k >= 0} binomial(2*k,k)*(2*x)^k/(1-x)^(k+1).
E.g.f.: exp(5*x)*BesselI(0, 4*x). (End)
a(n) = Sum_{k = 0..n} Sum_{j = 0..n-k} C(n,j)*C(n-j,k)*C(2*n-2*j,n-j). - Paul Barry, May 19 2006
a(n) = Sum_{k = 0..n} 4^k*C(n,k)^2. - heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010
a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Sep 11 2012
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - R. J. Mathar, Nov 26 2012
a(n) = hypergeom([-n, -n], [1], 4). - Vladimir Reshetnikov, Nov 29 2013
a(n) = hypergeom([-n, 1/2], [1], -8). - Peter Luschny, Apr 26 2016
From Michael Somos, Jun 01 2017: (Start)
a(n) = -3 * 9^n * a(-1-n) for all n in Z.
0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End)
From Peter Bala, Nov 13 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 5*x^2 + 29*x^3 + 185*x^4 + 1257*x^5 + ... is the g.f. of A059231.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all positive integers n and r and all primes p. (End)
a(n) = 3^n * LegendreP(n, 5/3). - G. C. Greubel, May 30 2023
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A047781 a(n) = Sum_{k=0..n-1} binomial(n-1,k)*binomial(n+k,k). Also a(n) = T(n,n), array T as in A049600.

Original entry on oeis.org

0, 1, 4, 19, 96, 501, 2668, 14407, 78592, 432073, 2390004, 13286043, 74160672, 415382397, 2333445468, 13141557519, 74174404608, 419472490257, 2376287945572, 13482186743203, 76598310928096, 435730007006341, 2481447593848524, 14146164790774359

Offset: 0

N. J. A. Sloane

nonn

Also main diagonal of array: m(i,1)=1, m(1,j)=j, m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 3 4 ... / 1 4 9 16 ... / 1 6 19 44 ... / 1 8 33 96 ... /. - Benoit Cloitre, Aug 05 2002
This array is now listed as A142978, where some conjectural congruences for the present sequence are given. - Peter Bala, Nov 13 2008
Convolution of central Delannoy numbers A001850 and little Schroeder numbers A001003. Hankel transform is 2^C(n+1,2)*A007052(n). - Paul Barry, Oct 07 2009
Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0 <= r <= n and the other terms recursively with T(r,c) = T(r-1,c-1) + 2*T(r,c-1). The sum of the last terms in the rows is Sum_{r=0..n} T(r,r) = a(n+1). Example: For n=4 the triangle has the rows 1; 4 9; 6 16 41; 4 14 44 129; 1 6 26 96 321 having sum of last terms 1 + 9 + 41 + 129 + 321 = 501 = a(5). - J. M. Bergot, Feb 15 2013
a(n) = A049600(2*n,n), when A049600 is seen as a triangle read by rows. - Reinhard Zumkeller, Apr 15 2014
a(n-1) for n > 1 is the number of assembly trees with the connected gluing rule for cycle graphs with n vertices. - Nick Mayers, Aug 16 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
A. Bacher, Directed and multi-directed animals on the square lattice with next nearest neighbor edges, arXiv preprint arXiv:1301.1365 [math.CO], 2013. See D(t). - From _N. J. A. Sloane_, Feb 14 2013
C. Banderier and P. Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. Table 2.
M. Bona and A. Vince, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
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.
A. Dougherty, N. Mayers, and R. Short, How to Build a Graph in n Days: Some Variants on Graph Assembly, arXiv preprint arXiv:1807.08079 [math.CO], 2018.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.
Milan Janjic, Two Enumerative Functions
G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

Cf. A002003. Column 1 of A296129.

a047781 n = a049600 (2 * n) n  -- Reinhard Zumkeller, Apr 15 2014

[n eq 0 select 0 else &+[Binomial(n-1, k)*Binomial(n+k, k): k in [0..n-1]]: n in [0..22]];  // Bruno Berselli, May 19 2011

a := proc(n) local k; add(binomial(n-1,k)*binomial(n+k,k),k=0..n-1); end;

Table[SeriesCoefficient[x*((1+x)-Sqrt[1-6*x+x^2])/(4*x*Sqrt[1-6*x+x^2]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[n_] := Hypergeometric2F1[1-n, n+1, 1, -1]; a[0] = 0; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 26 2013 *)
a[n_] := Sum[ Binomial[n - 1, k] Binomial[n + k, k], {k, 0, n - 1}]; Array[a, 25] (* Robert G. Wilson v, Aug 08 2018 *)

makelist(if n=0 then 0 else sum(binomial(n-1, k)*binomial(n+k, k), k, 0, n-1), n, 0, 22); /* Bruno Berselli, May 19 2011 */

A047781(n)=polcoeff((1+x)/sqrt(1+(O(x^n)-6)*x+x^2),n)\4  \\ M. F. Hasler, Oct 09 2012

from sympy import binomial
def a(n):
    return sum(binomial(n - 1, k) * binomial(n + k, k) for k in range(n))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 18 2017

from math import comb
def A047781(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

D-finite with recurrence n*(2*n-3)*a(n) - (12*n^2-24*n+8)*a(n-1) + (2*n-1)*(n-2)*a(n-2) = 0. - Vladeta Jovovic, Aug 29 2004
a(n+1) = Sum_{k=0..n} binomial(n, k)*binomial(n+1, k+1)*2^k. - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..n} T(n, k), array T as in A008288.
If shifted one place left, the third binomial transform of A098660. - Paul Barry, Sep 20 2004
G.f.: ((1+x)/sqrt(1-6x+x^2)-1)/4. - Paul Barry, Sep 20 2004, simplified by M. F. Hasler, Oct 09 2012
E.g.f. for sequence shifted left: Sum_{n>=0} a(n+1)*x^n/n! = exp(3*x)*(BesselI(0, 2*sqrt(2)*x)+BesselI(1, 2*sqrt(2)*x)/sqrt(2)). - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)*2^(n-k-1); a(n+1) = 2^n*Hypergeometric2F1(-n,-n-1;1;1/2). - Paul Barry, Feb 08 2011
a(n) ~ 2^(1/4)*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n >= 7. - Fung Lam, Feb 05 2014
a(n) = A241023(n) / 4. - Reinhard Zumkeller, Apr 15 2014
a(n) = Hyper2F1([-n, n], [1], -1)/2 for n > 0. - Peter Luschny, Aug 02 2014
n^2*a(n) = Sum_{k=0..n-1} (2*k^2+2*k+1)*binomial(n-1,k)*binomial(n+k,k). By the Zeilberger algorithm, both sides of the equality satisfy the same recurrence. - Zhi-Wei Sun, Aug 30 2014
a(n) = [x^n] (1/2) * ((1+x)/(1-x))^n for n > 0. - Seiichi Manyama, Jun 07 2018

A263159 Number A(n,k) of lattice paths starting at {n}^k and ending when k or any component equals 0, using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 15, 157, 63, 1, 1, 1, 31, 2101, 5419, 321, 1, 1, 1, 63, 32461, 717795, 220561, 1683, 1, 1, 1, 127, 580693, 142090291, 328504401, 9763807, 8989, 1, 1, 1, 255, 11917837, 39991899123, 944362553521, 172924236255, 454635973, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 11 2015

			Square array A(n,k) begins:
  1, 1,    1,       1,            1,                1, ...
  1, 1,    3,       7,           15,               31, ...
  1, 1,   13,     157,         2101,            32461, ...
  1, 1,   63,    5419,       717795,        142090291, ...
  1, 1,  321,  220561,    328504401,     944362553521, ...
  1, 1, 1683, 9763807, 172924236255, 7622403922836151, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A115866, A263162, A263163, A263164, A263165, A263166, A263167, A263168.
Rows n=0-1 give: A000012, A255047.
Main diagonal gives A263160.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A262809.

s:= proc(n) option remember; `if`(n=0, {[]},
      map(x-> [[x[], 0], [x[], 1]][], s(n-1)))
    end:
b:= proc(l) option remember; `if`(l=[] or l[1]=0, 1,
       add((p-> `if`(p[1]<0, 0, `if`(p[1]=0, 1, b(p)))
       )(sort(l-x)), x=s(nops(l)) minus {[0$nops(l)]}))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n,d-n), n=0..d), d=0..10);

g[k_] := Table[Reverse[IntegerDigits[n, 2]][[;;k]], {n, 2^k+1, 2^(k+1)-1}];
b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[k]}]];
a[n_, k_] := If[n == 0 || k == 0 || k == 1, 1, b[Table[n, {k}]]];
Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz in A115866 *)

A156289 Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 63, 210, 105, 1, 255, 2205, 3150, 945, 1, 1023, 21120, 65835, 51975, 10395, 1, 4095, 195195, 1201200, 1891890, 945945, 135135, 1, 16383, 1777230, 20585565, 58108050, 54864810, 18918900, 2027025, 1, 65535, 16076985

Offset: 1

Hartmut F. W. Hoft, Feb 07 2009

T(n,k) is the number of partitions of a set of size 2*n into k blocks of even size [Comtet]. For partitions into odd sized blocks see A136630.
See A241171 for the triangle of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. - Peter Bala, Aug 20 2014
This triangle T(n,k) gives the sum over the M_3 multinomials A036040 for the partitions of 2*n with k even parts, for 1 <= k <= n. See the triangle A257490 with sums over the entries with k parts, and the Hartmut F. W. Hoft program. - Wolfdieter Lang, May 13 2015

			The triangle begins
  n\k|..1.....2......3......4......5......6
  =========================================
  .1.|..1
  .2.|..1.....3
  .3.|..1....15.....15
  .4.|..1....63....210....105
  .5.|..1...255...2205...3150....945
  .6.|..1..1023..21120..65835..51975..10395
  ..
T(3,3) = 15. The 15 partitions of the set [6] into three even blocks are:
  (12)(34)(56), (12)(35)(46), (12)(36)(45),
  (13)(24)(56), (13)(25)(46), (13)(26)(45),
  (14)(23)(56), (14)(25)(36), (14)(26)(35),
  (15)(23)(46), (15)(24)(36), (15)(26)(34),
  (16)(23)(45), (16)(24)(35), (16)(25)(34).
Examples of recurrence relation
 T(4,3) = 5*T(3,2) + 9*T(3,3) = 5*15 + 9*15 = 210;
 T(6,5) = 9*T(5,4) + 25*T(5,5) = 9*3150 + 25*945 = 51975.
 T(4,2) = 28 + 35 = 63 (M_3 multinomials A036040 for partitions of 8 with 3 even parts, namely (2,6) and (4^2)). - _Wolfdieter Lang_, May 13 2015

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Richard Olu Awonusika, On Jacobi polynomials P_k(alpha, beta) and coefficients c_j^L(alpha, beta) (k >= 0, L = 5,6; 1 <= j <= L; alpha, beta > -1), The Journal of Analysis (2020).
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
J. Riordan, Letter, Jul 06 1978

Diagonal T(n, n) is A001147, subdiagonal T(n+1, n) is A001880.
2nd column variant T(n, 2)/3, for 2<=n, is A002450.
3rd column variant T(n, 3)/15, for 3<=n, is A002451.
Sum of the n-th row is A005046.
Cf. A241171, A257468, A257490, A096162.

T := proc(n,k) option remember; `if`(k = 0 and n = 0, 1, `if`(n < 0, 0,
(2*k-1)*T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 1 to 8 do seq(T(n,k), k=1..n) od; # Peter Luschny, Sep 04 2017

T[n_,k_] := Which[n < k, 0, n == 1, 1, True, 2/Factorial2[2 k] Sum[(-1)^(k + j) Binomial[2 k, k + j] j^(2 n), {j, 1, k}]]
(* alternate computation with function triangle[] defined in A257490 *)
a[n_]:=Map[Apply[Plus,#]&,triangle[n],{2}]
(* Hartmut F. W. Hoft, Apr 26 2015 *)

Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=n

Generating function for the k-th column of the triangle T(i+k,k):

G(k,x) = Sum_{i>=0} T(i+k,k)*x^i = Product_{j=1..k} (2*j-1)/(1-j^2*x).

Closed form expression: T(n,k) = (2/(k!*2^k))*Sum_{j=1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).

From Peter Bala, Feb 21 2011: (Start)

GENERATING FUNCTION

E.g.f. (including a constant 1):

(1)... F(x,z) = exp(x*(cosh(z)-1))

= Sum_{n>=0} R(n,x)*z^(2*n)/(2*n)!

= 1 + x*z^2/2! + (x + 3*x^2)*z^4/4! + (x + 15*x^2 + 15*x^3)*z^6/6! + ....

ROW POLYNOMIALS

The row polynomials R(n,x) begin

... R(1,x) = x

... R(2,x) = x + 3*x^2

... R(3,x) = x + 15*x^2 + 15*x^3.

The egf F(x,z) satisfies the partial differential equation

(2)... d^2/dz^2(F) = x*F + x*(2*x+1)*F' + x^2*F'',

where ' denotes differentiation with respect to x. Hence the row polynomials satisfy the recurrence relation

(3)... R(n+1,x) = x*{R(n,x) + (2*x+1)*R'(n,x) + x*R''(n,x)}

with R(0,x) = 1. The recurrence relation for T(n,k) given above follows from this.

(4)... T(n,k) = (2*k-1)!!*A036969(n,k).

(End)

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A036442 a(n) = 2^((n-1)*(n+2)/2).

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A049600 Array T read by diagonals; T(i,j) is the number of paths from (0,0) to (i,j) consisting of nonvertical segments (x(k),y(k))-to-(x(k+1),y(k+1)) such that 0 = x(1) < x(2) < ... < x(n-1) < x(n)=i, 0 = y(1) <= y(2) <= ... <= y(n-1) <= y(n)=j, for i >= 0, j >= 0.

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A120588 G.f. is 1 + x*c(x), where c(x) is the g.f. of the Catalan numbers (A000108).

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A069835 Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).

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A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

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A002003 a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k).

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.