A002003 -id:A002003 - cp's OEIS frontend

A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).

1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090

Offset: 0

N. J. A. Sloane

For the little Schröder numbers (or little Schroeder numbers, or small Schroeder numbers) see A001003.
The number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...). - Roberto E. Martinez II, Nov 05 2001
a(n) is the number of subdiagonal paths from (0, 0) to (n, n) consisting of steps East (1, 0), North (0, 1) and Northeast (1, 1) (sometimes called royal paths). - David Callan, Mar 14 2004
Twice A001003 (except for the first term).
a(n) is the number of dissections of a regular (n+4)-gon by diagonals that do not touch the base. (A diagonal is a straight line joining two nonconsecutive vertices and dissection means the diagonals are noncrossing though they may share an endpoint. One side of the (n+4)-gon is designated the base.) Example: a(1)=2 because a pentagon has only 2 such dissections: the empty one and the one with a diagonal parallel to the base. - David Callan, Aug 02 2004
a(n) is the number of separable permutations, i.e., permutations avoiding 2413 and 3142 (see Shapiro and Stephens). - Vincent Vatter, Aug 16 2006
Eric W. Weisstein comments that the Schröder numbers bear the same relationship to the Delannoy numbers (A001850) as the Catalan numbers (A000108) do to the binomial coefficients. - Jonathan Vos Post, Dec 23 2004
a(n) is the number of lattice paths from (0, 0) to (n+1, n+1) consisting of unit steps north N = (0, 1) and variable-length steps east E = (k, 0), with k a positive integer, that stay strictly below the line y = x except at the endpoints. For example, a(2) = 6 counts 111NNN, 21NNN, 3NNN, 12NNN, 11N1NN, 2N1NN (east steps indicated by their length). If the word "strictly" is replaced by "weakly", the counting sequence becomes the little Schröder numbers, A001003 (offset). - David Callan, Jun 07 2006
a(n) is the number of dissections of a regular (n+3)-gon with base AB that do not contain a triangle of the form ABP with BP a diagonal. Example: a(1) = 2 because the square D-C | | A-B has only 2 such dissections: the empty one and the one with the single diagonal AC (although this dissection contains the triangle ABC, BC is not a diagonal). - David Callan, Jul 14 2006
a(n) is the number of (colored) Motzkin n-paths with each upstep and each flatstep at ground level getting one of 2 colors and each flatstep not at ground level getting one of 3 colors. Example: With their colors immediately following upsteps/flatsteps, a(2) = 6 counts U1D, U2D, F1F1, F1F2, F2F1, F2F2. - David Callan, Aug 16 2006
The Hankel transform of this sequence is A006125(n+1) = [1, 2, 8, 64, 1024, 32768, ...]; example: Det([1, 2, 6, 22; 2, 6, 22, 90; 6, 22, 90, 394; 22, 90, 394, 1806]) = 64. - Philippe Deléham, Sep 03 2006
Triangle A144156 has row sums equal to A006318 with left border A001003. - Gary W. Adamson, Sep 12 2008
a(n) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain). Equivalently, it is the order of the Schröder monoid, PC sub n. - Abdullahi Umar, Oct 02 2008
Sum_{n >= 0} a(n)/10^n - 1 = (9 - sqrt(41))/2. - Mark Dols, Jun 22 2010
1/sqrt(41) = Sum_{n >= 0} Delannoy number(n)/10^n. - Mark Dols, Jun 22 2010
a(n) is also the dimension of the space Hoch(n) related to Hochschild two-cocycles. - Ph. Leroux (ph_ler_math(AT)yahoo.com), Aug 24 2010
Let W = (w(n, k)) denote the augmentation triangle (as at A193091) of A154325; then w(n, n) = A006318(n). - Clark Kimberling, Jul 30 2011
Conjecture: For each n > 2, the polynomial sum_{k = 0}^n a(k)*x^{n-k} is irreducible modulo some prime p < n*(n+1). - Zhi-Wei Sun, Apr 07 2013
From Jon Perry, May 24 2013: (Start)
Consider a Pascal triangle variant where T(n, k) = T(n, k-1) + T(n-1, k-1) + T(n-1, k), i.e., the order of performing the calculation must go from left to right (A033877). This sequence is the rightmost diagonal.
Triangle begins:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 16, 22;
  1,  8, 30, 68, 90;
  ... (End)
a(n) is the number of permutations avoiding 2143, 3142 and one of the patterns among 246135, 254613, 263514, 524361, 546132. - Alexander Burstein, Oct 05 2014
a(n) is the number of semi-standard Young tableaux of shape n x 2 with consecutive entries.  That is, j in P and 1 <= i<= j imply i in P. - Graham H. Hawkes, Feb 15 2015
a(n) is the number of unary-rooted size n unary-binary trees (each node has either 1 or 2 degree out). - John Bodeen, May 29 2017
Conjecturally, a(n) is the number of permutations pi of length n such that s(pi) avoids the patterns 231 and 321, where s denotes West's stack-sorting map. - Colin Defant, Sep 17 2018
a(n) is the number of n X n permutation matrices which percolate under the 2-neighbor bootstrap percolation rule (see Shapiro and Stephens). The number of general n X n matrices of weight n which percolate is given in A146971. - Jonathan Noel, Oct 05 2018
a(n) is the number of permutations of length n+1 which avoid 3142 and 3241. The permutations are precisely the permutations that are sortable by a decreasing stack followed by an increasing stack in series. - Rebecca Smith, Jun 06 2019
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {3>1, 4>1, 1>2} of length 4. That is, the number of length n+1 permutations having no subsequences of length 4 in which the second element is the smallest, and the first element is smaller than the third and fourth elements. - Sergey Kitaev, Dec 10 2020
Named after the German mathematician Ernst Schröder (1841-1902). - Amiram Eldar, Apr 15 2021
a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_i <= i for all i, and (ii) the nonzero u_i are weakly increasing. For example, a(2) = 6 counts 00, 01, 02, 10, 11, 12. See link "Some bijections for lattice paths" at A001003. - David Callan, Dec 18 2021
a(n) is the number of separable elements of the Weyl group of type B_n/C_n (see Gaetz and Gao). - Fern Gossow, Jul 31 2023
The number of domino tilings of an Aztec triangle of order n. Dually, the number perfect matchings of the edges in the cellular graph formed by a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...) as in Ciucu (1996). - Michael Somos, Sep 16 2024
a(n) is the number of dissections of a convex (n+3)-sided polygon by non-intersecting diagonals such that none of the dividing diagonals passes through a chosen vertex. - Muhammed Sefa Saydam, Mar 01 2025
a(n) is the number of dissections of a convex (n+m+1)-sided polygon by non-intersecting diagonals such that the selected m consecutive sides of the polygon will be in the same subpolygon. - Muhammed Sefa Saydam, Jul 02 2025

			a(3) = 22 since the top row of Q^n = (6, 6, 6, 4, 0, 0, 0, ...); where 22 = (6 + 6 + 6 + 4).
G.f. = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + 1806*x^6 + 8858*x^7 + 41586*x^8 + ...

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P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
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Feng Qi and B.-N. Guo, Some explicit and recursive formulas of the large and little Schröder numbers, Arab Journal of Mathematical Sciences, June 2016.
Feng Qi, Xiao-ting Shi, and Bai-Ni Guo, Integral representations of the large and little Schroder numbers, preprint, 2016.
Markus Saers, Dekai Wu, and Chris Quirk, On the Expressivity of Linear Transductions, The 13th Machine Translation Summit.
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Index entries for "core" sequences

Apart from leading term, twice A001003 (the small Schroeder numbers). Cf. A025240.
Sequences A085403, A086456, A103137, A112478 are essentially the same sequence.
Main diagonal of A033877.
Cf. A002003, A004148, A088617, A060693, A144156.
Row sums of A104219. Bisections give A138462, A138463.
Row sums of A175124.
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

Concatenation([1],List([1..25],n->(1/n)*Sum([0..n],k->2^k*Binomial(n,k)*Binomial(n,k-1)))); # Muniru A Asiru, Nov 29 2018

a006318 n = a004148_list !! n
a006318_list = 1 : f [1] where
   f xs = y : f (y : xs) where
     y = head xs + sum (zipWith (*) xs $ reverse xs)
-- Reinhard Zumkeller, Nov 13 2012

Order := 24: solve(series((y-y^2)/(1+y),y)=x,y); # then A(x)=y(x)/x
BB:=(-1-z-sqrt(1-6*z+z^2))/2: BBser:=series(BB, z=0, 24): seq(coeff(BBser, z, n), n=1..23); # Zerinvary Lajos, Apr 10 2007
A006318_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A006318_list(22); # Peter Luschny, May 19 2011
A006318 := n-> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A006318(n), n=0..22); # Johannes W. Meijer, Jul 14 2013
seq(simplify(hypergeom([-n,n+1],[2],-1)), n=0..100); # Robert Israel, Mar 23 2015

a[0] = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k]*a[n - 1 - k], {k, 0, n - 1}]; Array[a[#] &, 30]
InverseSeries[Series[(y - y^2)/(1 + y), {y, 0, 24}], x] (* then A(x) = y(x)/x *) (* Len Smiley, Apr 11 2000 *)
CoefficientList[Series[(1 - x - (1 - 6x + x^2)^(1/2))/(2x), {x, 0, 30}], x] (* Harvey P. Dale, May 01 2011 *)
a[ n_] := 2 Hypergeometric2F1[ -n + 1, n + 2, 2, -1]; (* Michael Somos, Apr 03 2013 *)
a[ n_] := With[{m = If[ n < 0, -1 - n, n]}, SeriesCoefficient[(1 - x - Sqrt[ 1 - 6 x + x^2])/(2 x), {x, 0, m}]]; (* Michael Somos, Jun 10 2015 *)
Table[-(GegenbauerC[n+1, -1/2, 3] + KroneckerDelta[n])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Nov 12 2016 *)
CoefficientList[Nest[1+x(#+#^2)&, 1+O[x], 20], x] (* Oliver Seipel, Dec 21 2024 *)

{a(n) = if( n<0, n = -1-n); polcoeff( (1 - x - sqrt( 1 - 6*x + x^2 + x^2 * O(x^n))) / 2, n+1)}; /* Michael Somos, Apr 03 2013 */

{a(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n)};

from gmpy2 import divexact
A006318 = [1, 2]
for n in range(3,10**3):
    A006318.append(int(divexact(A006318[-1]*(6*n-9)-(n-3)*A006318[-2],n)))
# Chai Wah Wu, Sep 01 2014

# Generalized algorithm of L. Seidel
def A006318_list(n) :
    D = [0]*(n+1); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1;
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1]);
        b = not b
    return R
A006318_list(23) # Peter Luschny, Jun 02 2012

G.f.: (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x).
a(n) = 2*hypergeom([-n+1, n+2], [2], -1). - Vladeta Jovovic, Apr 24 2003
For n > 0, a(n) = (1/n)*Sum_{k = 0..n} 2^k*C(n, k)*C(n, k-1). - Benoit Cloitre, May 10 2003
The g.f. satisfies (1 - x)*A(x) - x*A(x)^2 = 1. - Ralf Stephan, Jun 30 2003
For the asymptotic behavior, see A001003 (remembering that A006318 = 2*A001003). - N. J. A. Sloane, Apr 10 2011
From Philippe Deléham, Nov 28 2003: (Start)
Row sums of A088617 and A060693.
a(n) = Sum_{k = 0..n} C(n+k, n)*C(n, k)/(k+1). (End)
With offset 1: a(1) = 1, a(n) = a(n-1) + Sum_{i = 1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k = 0..n} A000108(k)*binomial(n+k, n-k). - Benoit Cloitre, May 09 2004
a(n) = Sum_{k = 0..n} A011117(n, k). - Philippe Deléham, Jul 10 2004
a(n) = (CentralDelannoy(n+1) - 3 * CentralDelannoy(n))/(2*n) = (-CentralDelannoy(n+1) + 6 * CentralDelannoy(n) - CentralDelannoy(n-1))/2 for n >= 1, where CentralDelannoy is A001850. - David Callan, Aug 16 2006
From Abdullahi Umar, Oct 11 2008: (Start)
A123164(n+1) - A123164(n) = (2*n+1)*a(n) (n >= 0).
  and 2*A123164(n) = (n+1)*a(n) - (n-1)*a(n-1) (n > 0). (End)
Define the general Delannoy numbers d(i, j) as in A001850. Then a(k) = d(2*k, k) - d(2*k, k-1) and a(0) = 1, Sum_{j=0..n} ((-1)^j * (d(n, j) + d(n-1, j-1)) * a(n-j)) = 0. - Peter E John, Oct 19 2006
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([2]). - Gary W. Adamson, Oct 27 2008
G.f.: 1/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x.... (continued fraction). - Paul Barry, Dec 08 2008
G.f.: 1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - Paul Barry, Jan 29 2009
a(n) ~ ((3 + 2*sqrt(2))^n)/(n*sqrt(2*Pi*n)*sqrt(3*sqrt(2) - 4))*(1-(9*sqrt(2) + 24)/(32*n) + ...). - G. Nemes (nemesgery(AT)gmail.com), Jan 25 2009
Logarithmic derivative yields A002003. - Paul D. Hanna, Oct 25 2010
a(n) = the upper left term in M^(n+1), M = the production matrix:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  4, 4, 2, 1, 1, 0, ...
  8, 8, 8, 2, 1, 1, ...
  ... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms in Q^n, Q = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 2, 0, 0, 0, ...
  1, 1, 1, 2, 0, 0, ...
  1, 1, 1, 1, 2, 0, ...
  1, 1, 1, 1, 1, 2, ...
  ... - Gary W. Adamson, Aug 23 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x) = (1 - 3*x - sqrt(1 - 6*x + x^2))/(2*x) an o.g.f. (nulling the n = 0 term) for A006318, G(x) = x/(2 + 3*x + x^2) is the compositional inverse.
Consequently, with H(x) = 1/ (dG(x)/dx) = (2 + 3*x + x^2)^2 / (2 - x^2),
  a(n) = (1/n!)*[(H(x)*d/dx)^n] x evaluated at x = 0, i.e.,
  F(x) = exp[x*H(u)*d/du] u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)). (End)
a(n-1) = number of ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n - 1 - k internal vertices colored white, and such that each vertex and its rightmost child have different colors ([Drake, Example 1.6.7]). For a refinement of this sequence see A175124. - Peter Bala, Sep 29 2011
D-finite with recurrence: (n-2)*a(n-2) - 3*(2*n-1)*a(n-1) + (n+1)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (1 - G(0))/x; G(k) = 1 + x - 2*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (G(0) - 1)/x; G(k) = 1 - x/(1 - 2/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jan 04 2012
a(n+1) = a(n) + Sum_{k=0..n} a(k)*(n-k). - Reinhard Zumkeller, Nov 13 2012
G.f.: 1/Q(0) where Q(k) = 1 + k*(1 - x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(-1-n) = a(n). - Michael Somos, Apr 03 2013
G.f.: 1/x - 1 - U(0)/x, where U(k) = 1 - x - x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
G.f.: (2 - 2*x - G(0))/(4*x), where G(k) = 1 + 1/( 1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) = 1/(n + 1) * (Sum_{j=0..n} C(n+j, j)*C(n+j+1, j+1)*(Sum_{k=0..n-j} (-1)^k*C(n+j+k, k))). - Graham H. Hawkes, Feb 15 2015
a(n) = hypergeom([-n, n+1], [2], -1). - Peter Luschny, Mar 23 2015
a(n) = sqrt(2) * LegendreP(n, -1, 3) where LegendreP is the associated Legendre function of the first kind (in Maple's notation). - Robert Israel, Mar 23 2015
G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*A(x)^k. - Ilya Gutkovskiy, Apr 11 2019
From Peter Bala, May 13 2024: (Start)
a(n) = 2 * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*binomial(2*n-2*k, n)/(n-2*k+1) for n >= 1.
a(n) = Integral_{x = 0..1} Legendre_P(n, 2*x+1) dx. (End)
G.f. A(x) = 1/(1 - x) * c(x/(1-x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Aug 29 2024

Edited by Charles R Greathouse IV, Apr 20 2010

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

A035607 Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 6, 8, 2, 1, 8, 18, 12, 2, 1, 10, 32, 38, 16, 2, 1, 12, 50, 88, 66, 20, 2, 1, 14, 72, 170, 192, 102, 24, 2, 1, 16, 98, 292, 450, 360, 146, 28, 2, 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2, 1, 20, 162, 688, 1666, 2364, 1970, 952, 258, 36, 2, 1, 22, 200, 978, 2816

Offset: 0

N. J. A. Sloane

Table also gives coordination sequences of same lattices.
Rows sums are given by A001333. Rising and falling diagonals are the tribonacci numbers A000213, A001590. - Paul Barry, Feb 13 2003
a(d,m) also gives the number of ways to choose m squares from a 2 X (d-1) grid so that no two squares in the selection are (horizontally or vertically) adjacent. - Jacob A. Siehler, May 13 2006
Mirror image of triangle A113413. - Philippe Deléham, Oct 15 2006
The Ca1 sums lead to A126116 and the Ca2 sums lead to A070550, see A180662 for the definitions of these triangle sums. - Johannes W. Meijer, Aug 05 2011
A035607 is jointly generated with the Delannoy triangle A008288 as an array of coefficients of polynomials v(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 05 2012
Also, the polynomial v(n,x) above is x + (x + 1)*f(n-1,x), where f(0,x) = 1. - Clark Kimberling, Oct 24 2014
Rows also give the coefficients of the independence polynomial of the n-ladder graph. - Eric W. Weisstein, Dec 29 2017
Considering both sequences as square arrays (offset by one row), the rows of A035607 are the first differences of the rows of A008288, and the rows of A008288 are the partial sums of the rows of A035607. - Shel Kaphan, Feb 23 2023
Considering only points with nonnegative coordinates, the number of points at L1 distance = m in d dimensions is the same as the number of ways of putting m indistinguishable balls into d distinguishable urns, binomial(m+d-1, d-1). This is one facet of the cross-polytope. Allowing for + and - coordinates, there are binomial(d,i)*2^i facets containing points with up to i nonzero coordinates. Eliminating double counting of points with any coordinates = 0, there are Sum_{i=1..d} (-1)^(d-i)*binomial(m+i-1,i-1)*binomial(d,i)*2^i points at distance m in d dimensions. One may avoid the alternating sum by using binomial(m-1,i-1) to count only the points per facet with exactly i nonzero coordinates, avoiding any double counting, but the result is the same. - Shel Kaphan, Mar 04 2023

			From _Clark Kimberling_, Oct 24 2014: (Start)
As a triangle of coefficients in polynomials v(n,x) in Comments, the first 6 rows are
  1
  1   2
  1   4   2
  1   6   8   2
  1   8  18  12   2
  1  10  32  38  16   2
  ... (End)
From _Shel Kaphan_, Mar 04 2023: (Start)
For d=3, m=4:
There are binomial(3,1)*2^1 = 6 facets (vertices) of binomial(4+1-1,1-1) = 1 point with <= one nonzero coordinate.
There are binomial(3,2)*2^2 = 12 facets (edges) of binomial(4+2-1,2-1) = 5 points with <= two nonzero coordinates.
There are binomial(3,3)*2^3 = 8 facets (faces) of binomial(4+3-1,3-1) = 15 points with <= three nonzero coordinates.
a(3,4) = 8*15 - 12*5 + 6*1 = 120 - 60 + 6 = 66. (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 392.
R. J. Mathar, Bivariate Generating Functions for Non-attacking Wazirs on Rectangular Boards (2024), Table 2.
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
J. Siehler, Adjacency-free selections from a 2xN grid (Mathematica notebook) [broken link]
Jacob A. Siehler, Selections Without Adjacency on a Rectangular Grid, arXiv:1409.3869 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Ladder Graph

Other versions: A113413, A119800, A122542, A266213.
Cf. A008288, which has g.f. 1/(1-x-x*y-x^2*y).
Cf. A078057 (row sums), A050146 (central terms).
Cf. A050146.

a035607 n k = a035607_tabl !! n !! k
a035607_row n = a035607_tabl !! n
a035607_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 2])
-- Reinhard Zumkeller, Jul 20 2013

A035607 := proc(d,m) local j: add(binomial(floor((d-1+j)/2),d-m-1)*binomial(d-m-1, floor((d-1-j)/2)),j=0..d-1) end: seq(seq(A035607(d,m),m=0..d-1),d=1..11); # d=dimension, m=norm # Johannes W. Meijer, Aug 05 2011

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 *)
Reverse /@ CoefficientList[CoefficientList[Series[(1 + x)/(1 - x - x y - x^2 y), {x, 0, 10}], x], y] // Flatten (* Eric W. Weisstein, Dec 29 2017 *)

T(n, k) = if (k==0, 1, sum(i=0, k-1, binomial(n-k,i+1)*binomial(k-1,i)*2^(i+1)));
tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(T(n, k), ", ")); print); \\ as a triangle; Michel Marcus, Feb 27 2018

def A035607_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-i,k-1) for i in (2..n-k+1))
    return [prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A035607_row(n)) # Peter Luschny, Mar 16 2016

From Johannes W. Meijer, Aug 05 2011: (Start)
f(d,m) = Sum_{j=0..d-1} binomial(floor((d-1+j)/2), d-m-1)*binomial(d-m-1, floor((d-1-j)/2)), d >= 1 and 0 <= m <= d-1.
f(d,m) = f(d-1,m-1) + f(d-1,m) + f(d-2,m-1) (d >= 3 and 1 <= m <= d-1) with f(d,0) = 1 (d >= 1) and f(d,d-1) = 2 (d>=2). (End)
From Roger Cuculière, Apr 10 2006: (Start)
The generating function G(x,y) of this double sequence is the sum of a(n,p)*x^n*y^p, n=1..oo, p=0..oo, which is G(x,y) = x*(1+y)/(1-x-y-x*y).
The horizontal generating function H_n(y), which generates the rows of the table: (1, 2, 2, 2, 2, ...), (1, 4, 8, 12, 16, ...), (1, 6, 18, 38, 66, ...), is the sum of a(n,p)*y^p, p=0..oo, for each fixed n. This is H_n(y) = ((1+y)^n)/((1-y)^n).
The vertical generating function V_p(x), which generates the columns of the table: (1, 1, 1, 1, 1, ...), (2, 4, 6, 8, 10, ...), (2, 8, 18, 32, 50, ...), is the sum of a(n,p)*x^n, n=1..oo, for each fixed p. This is V_p(x) = 2*((1+x)^(p-1))/((1-x)^(p+1)) for p >= 1 and V_0(x) = x/(1-x). (End)
G.f.: (1+x)/(1-x-x*y-x^2*y). - Vladeta Jovovic, Apr 02 2002 (But see previous lines!)
T(2*n,n) = A050146(n+1). - Reinhard Zumkeller, Jul 20 2013
Seen as a triangle read by rows: T(n,0) = 1, for n > 1: T(n,n-1) = 2, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1), 0 < k < n. - Reinhard Zumkeller, Jul 20 2013
Seen as a triangle T(n,k) with 0 <= k < n read by rows: T(n,0)=1 for n > 0 and T(n,k) = Sum_{i=0..k-1} binomial(n-k,i+1)*binomial(k-1,i)*2^(i+1) for k > 0. - Werner Schulte, Feb 22 2018
With p >= 1 and q >= 0, as a square array a(p,q) = T(p+q-1,q) = 2*p*Hypergeometric2F1[1-p, 1-q, 2, 2] for q >= 1. Consequently, a(p,q) = a(q,p)*p/q. - Shel Kaphan, Feb 14 2023
For n >= 1, T(2*n,n) = A002003(n), T(3*n,2*n) = A103885(n) and T(4*n,3*n) = A333715(n). - Peter Bala, Jun 15 2023

More terms from David W. Wilson

Maple program corrected and information added by Johannes W. Meijer, Aug 05 2011

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

A103885 a(n) = [x^(2*n)] ((1 + x)/(1 - x))^n.

Original entry on oeis.org

1, 2, 16, 146, 1408, 14002, 142000, 1459810, 15158272, 158611106, 1669752016, 17664712562, 187641279616, 2000029880786, 21380213588848, 229129634462146, 2460955893981184, 26482855453375042, 285475524009208720, 3082024598888203090, 33319523640218177408

Offset: 0

Ralf Stephan, Feb 20 2005

From Peter Bala, Mar 01 2020: (Start)
The recurrence given below can be rewritten in the form
(2*n+1)*(2*n+2)*P(2,n)*a(n+1) - (2*n-1)*(2*n-2)*P(2,-n)*a(n-1) = Q(2,n^2)*a(n), where the polynomial Q(2,n) = 4*(55*n^2 - 34*n + 3) and the polynomial P(2,n) = 5*n^2 - 5*n + 1 satisfies the symmetry condition P(2,n) = P(2,1-n) and has real zeros.
More generally, for fixed m = 1,2,3,..., we conjecture that the sequence b(n) := a(m*n) satisfies a recurrence of the form ( Product_{k = 1..2*m} (2*m*n + k) ) * P(2*m,n)*b(n+1) + (-1)^m*( Product_{k = 1..2*m} (2*m*n - k) ) * P(2*m,-n)*b(n-1) = Q(2*m,n^2)*b(n), where the polynomials P(2*m,n) and Q(2*m,n) have degree 2*m. Conjecturally, the polynomial P(2*m,n) = P(2*m,1-n) and has real zeros in the interval [0, 1]. The 4*m zeros of the polynomial Q(2*m,n^2) seem to belong to the interval [-1, 1] and 4*m - 2 of these zeros appear to be approximated by the rational numbers +- k/(3*m), where 1 <= k <= 3*m - 2, k not a multiple of 3. (End)

G. C. Greubel, Table of n, a(n) for n = 0..950 [a(0) = 1 inserted by _Georg Fischer_, Apr 03 2020]
Peter Bala, Notes on A103885
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A002003, A006318, A027307, A103882, A103884, A123164, A144097, A156894, A266213, A370102.

A103885:= func< n | n eq 0 select 1 else (&+[ Binomial(n, k)*Binomial(2*n+k-1, n-1): k in [0..n]]) >;
[A103885(n): n in [0..40]]; // G. C. Greubel, Oct 27 2024

a := n -> `if`(n=0, 1, 2*n*hypergeom([1 - 2*n, 1 - n], [2], 2)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Dec 30 2019
# Alternative (after Peter Bala ):
gf := n -> ( (1 + x)/(1 - x) )^n: ser := n -> series(gf(n), x, 40):
seq(coeff(ser(n), x, 2*n), n=0..17); # Peter Luschny, Mar 20 2020

Prepend[Table[Sum[2^i Binomial[n, i] Binomial[2n-1, i-1], {i, 1, 2n}], {n,1,20}], 1] (* Vaclav Kotesovec, Jul 01 2015 *)

a(n) = if (n==0, 1, sum(i=0, n, 2^i * binomial(n, i) * binomial(2*n-1, i-1))); \\ Michel Marcus, Mar 21 2020

def A103885(n): return 1 if n==0 else sum(binomial(n, k)*binomial(2*n+k-1, n-1) for k in range(n+1))
[A103885(n) for n in range(41)] # G. C. Greubel, Oct 27 2024

a(n) = Sum_{i=0..n} 2^i * binomial(n,i) * binomial(2*n-1,i-1). [Original definition, with summation range {i=1..n}.]
a(n) = A103884(n, n).
G.f.: A(x) = x*B(x)'/B(x), where B(x) is g.f. of A027307. - Vladimir Kruchinin, Jun 30 2015
From Vaclav Kotesovec, Jul 01 2015: (Start)
Recurrence: n*(2*n-1)*(5*n^2 - 15*n + 11)*a(n) = 2*(55*n^4 - 220*n^3 + 296*n^2 - 152*n + 24)*a(n-1) + (n-2)*(2*n-3)*(5*n^2 - 5*n + 1)*a(n-2).
a(n) ~ ((11 + 5*sqrt(5))/2)^n / (2 * 5^(1/4) * sqrt(Pi*n)). (End)
a(n) = [x^n] (1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ...))))))^n, a continued fraction. - Ilya Gutkovskiy, Sep 29 2017
a(n) = 2*n*hypergeom([1 - 2*n, 1 - n], [2], 2) for n >= 1. - Peter Luschny, Dec 30 2019
From Peter Bala, Mar 01 2020: (Start)
a(n) = Sum_{k = 0..n} C(n, k)*C(2*n+k-1, n-1), with a(0) = 1.
a(n) = Sum_{k = 0..n} C(2*n, 2*k)*C(2*n-k-1, n-1), with a(0) = 1.
a(n) = (1/2)*Sum_{k = 0..n} C(2*n, n-k)*C(2*n+k-1, k). Cf. A156894.
a(n) = [x^n] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the sequence of large Schröder numbers A006318.
a(n) = (1/2) * [x^(n)] ( (1 + x)/(1 - x) )^(2*n). Cf. A002003(n) = [x^n] ( (1 + x)/(1 - x) )^n.
Conjecture: a(n) = - [x^n] G(x)^(-n), where G(x) = 1 + 2*x + 14*x^2 + 134*x^3 + 1482*x^4 + ... is the o.g.f. of A144097.
a(p) == 2 ( mod p^3 ) for prime p >= 5. (End)
From Peter Bala, Sep 22 2021: (Start)
a(n) = Sum_{k = 0..n} 4^k*binomial(n+k-1,n)*binomial(n,k)^2 / binomial(2*k,k).
Equivalently, a(n) = [x^n] T(n,(1+x)/(1-x)), where T(n,x) is the n-th Chebyshev polynomial of the first kind. Cf. A103882. (End)
For n>0, a(n) = (1/3) * [x^n] (1/S(-x))^(3*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the sequence of large Schröder numbers A006318. Cf. A370102. - Peter Bala, Jul 29 2024

a(0) = 1 added and new name by Peter Bala, Mar 01 2020

A119259 Central terms of the triangle in A119258.

Original entry on oeis.org

1, 3, 17, 111, 769, 5503, 40193, 297727, 2228225, 16807935, 127574017, 973168639, 7454392321, 57298911231, 441739706369, 3414246490111, 26447737520129, 205272288591871, 1595964714385409, 12427568655368191, 96905907580960769, 756583504975757311, 5913649000782757889

Offset: 0

Reinhard Zumkeller, May 11 2006

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 06 2022

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, eq (42).

Cf. A005408, A056220, A000225, A000337, A055580, A027608, A119258, A064062, A178792, A014300, A098665.

a119259 n = a119258 (2 * n) n  -- Reinhard Zumkeller, Aug 06 2014

Table[Binomial[2k - 1, k] Hypergeometric2F1[-2k, -k, 1 - 2k, -1], {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

from itertools import count, islice
def A119259_gen(): # generator of terms
    yield from (1,3)
    a, c = 2, 1
    for n in count(1):
        yield (a<>1
A119259_list = list(islice(A119259_gen(),20)) # Chai Wah Wu, Apr 26 2023

a(n) = A119258(2*n,n).
a(n) = Sum_{k=0..n} C(2*n,k)*C(2*n-k-1,n-k). - Paul Barry, Sep 28 2007
a(n) = Sum_{k=0..n} C(n+k-1,k)*2^k. - Paul Barry, Sep 28 2007
2*a(n) = A064062(n)+A178792(n). - Joseph Abate, Jul 21 2010
G.f.: (4*x^2+3*sqrt(1-8*x)*x-5*x)/(sqrt(1-8*x)*(2*x^2+x-1)-8*x^2-7*x+1). - Vladimir Kruchinin, Aug 19 2013
a(n) = (-1)^n - 2^(n+1)*binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2],2). - Peter Luschny, Jul 25 2014
a(n) = (-1)^n + 2^(n+1)*A014300(n). - Peter Luschny, Jul 25 2014
a(n) = [x^n] ( (1 + x)^2/(1 - x) )^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 13*x^2 + 67*x^3 + ... is essentially the o.g.f. for A064062. - Peter Bala, Oct 01 2015
The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^2/(1 - x)) and hence is algebraic by Stanley 1999, Theorem 6.33, p.197. - Peter Bala, Aug 21 2016
n*(3*n-4)*a(n) +(-21*n^2+40*n-12)*a(n-1) -4*(3*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017
From Peter Bala, Mar 23 2020: (Start)
a(p) == 3 ( mod p^3 ) for prime p >= 5. Cf. A002003, A103885 and A156894.
More generally, we conjecture that a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. (End)
G.f.: (8*x)/(sqrt(1-8*x)*(1+4*x)-1+8*x). - Fabian Pereyra, Jul 20 2024
a(n) = 2^(n+1)*binomial(2*n,n) - A178792(n). - Akiva Weinberger, Dec 06 2024
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n,k). - Seiichi Manyama, Jul 31 2025

A050146 a(n) = T(n,n), array T as in A050143.

Original entry on oeis.org

1, 1, 4, 18, 88, 450, 2364, 12642, 68464, 374274, 2060980, 11414898, 63521352, 354870594, 1989102444, 11180805570, 63001648608, 355761664002, 2012724468324, 11406058224594, 64734486343480, 367891005738690, 2093292414443164, 11923933134635298, 67990160422313808

Offset: 0

Clark Kimberling

nonn

Also main diagonal of array : m(i,1)=1, i>=1; m(1,j)=2, j>1; m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 2 2 ... / 1 4 8 12 ... / 1 6 18 38 ... / 1 8 32 88 ... / - Benoit Cloitre, Aug 05 2002
a(n) is also the number of order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Aug 25 2008
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,c-1) + 2*T(r-1,c-1). The sum of the last terms in each row is Sum_{r=0..n} T(r,r)=a(n+1). For n=4 the triangle is 1; 4 6; 6 14 26; 4 16 44 96; 1 9 41 129 321 with the sum of the last terms being 1 + 6 + 26 + 96 + 321 = 450 = a(5). - J. M. Bergot, Jan 29 2013
It may be better to define a(0) = 0 for formulas without exceptions. - Michael Somos, Nov 25 2016
a(n) is the number of points at L1 distance n-1 from any point in Z^n, for n>=1. - Shel Kaphan, Mar 24 2023

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 450*x^5 + 2364*x^6 + 12642*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Cf. A002003, A006318, A050151, A008288, A002002, A026002.

-1-diagonal of A266213 for n>=1.

a050146 n = if n == 0 then 1 else a035607 (2 * n - 2) (n - 1)
-- Reinhard Zumkeller, Nov 05 2013, Jul 20 2013

Flatten[{1,RecurrenceTable[{(n-3)*(n-1)*a[n-2]-3*(n-2)*(2*n-3)*a[n-1]+(n-2)*(n-1)*a[n]==0,a[1]==1,a[2]==4},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_] := If[ n == 0, 1, Sum[ Binomial[n, k] Binomial[n + k - 2, k - 1], {k, n}]]; (* Michael Somos, Nov 25 2016 *)
a[ n_] := If[ n == 0, 1, n Hypergeometric2F1[1 - n, n, 2, -1]]; (* Michael Somos, Nov 25 2016 *)

taylor(-(x^4+sqrt(x^2-6*x+1)*(x^3-5*x^2+5*x+1)-8*x^3+16*x^2-6*x+1)/(x^3+sqrt(x^2-6*x+1)*(x^2-4*x-1)-7*x^2+7*x-1),x,0,10); /* Vladimir Kruchinin, Nov 25 2016 */

a(n)=if(n==0, 1, sum(k=1,n, binomial(n, k)*binomial(n+k-2, k-1)) ); \\ Joerg Arndt, May 04 2013

A050146 = lambda n : n*hypergeometric([1-n, n], [2], -1) if n>0 else 1
[round(A050146(n).n(100)) for n in (0..24)] # Peter Luschny, Sep 17 2014

From Vladeta Jovovic, Mar 31 2004: (Start)
Coefficient of x^(n-1) in expansion of ((1+x)/(1-x))^n, n > 0.
a(n) = Sum_{k=1..n} binomial(n, k)*binomial(n+k-2, k-1), n > 0. (End)
D-finite with recurrence (n-1)*(n-2)*a(n) = 3*(2*n-3)*(n-2)*a(n-1) - (n-1)*(n-3)*a(n-2) for n > 2. - Vladeta Jovovic, Jul 16 2004
a(n+1) = Jacobi_P(n, 1, -1, 3); a(n+1) = Sum{k=0..n} C(n+1, k)*C(n-1, n-k)*2^k. - Paul Barry, Jan 23 2006
a(n) = n*A006318(n-1) - Abdullahi Umar, Aug 25 2008
a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
a(n+1) = A035607(2*n,n). - Reinhard Zumkeller, Jul 20 2013
a(n) = n*hypergeometric([1-n, n], [2], -1) for n >= 1. - Peter Luschny, Sep 17 2014
O.g.f.: -(x^4 + sqrt(x^2 - 6*x + 1)*(x^3 - 5*x^2 + 5*x + 1) - 8*x^3 + 16*x^2 - 6*x + 1)/(x^3 + sqrt(x^2 - 6*x + 1)*(x^2 - 4*x - 1)- 7*x^2 + 7*x - 1). - Vladimir Kruchinin, Nov 25 2016
0 = a(n)*(a(n+1) - 18*a(n+2) + 65*a(n+3) - 12*a(n+4)) + a(n+1)*(54*a(n+2) - 408*a(n+3) + 81*a(n+4)) + a(n+2)*(72*a(n+2) + 334*a(n+3) - 90*a(n+4)) + a(n+3)*(-24*a(n+3) + 9*a(n+4)) for all integer n if a(0) = 0 and a(n) = -2*A050151(-n) for n < 0. - Michael Somos, Nov 25 2016
O.g.f: (2 - x + x*(3 - x)/sqrt(x^2 - 6*x + 1))/2. - Petros Hadjicostas, Feb 14 2021
a(n) = A002002(n) - A026002(n-1) for n>=2. - Shel Kaphan, Mar 24 2023

A123164 Row sums of A123160.

Original entry on oeis.org

1, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718, 161439727075246592

Offset: 0

Roger L. Bagula, Oct 02 2006

nonn

Coefficient of x^n in ((1 + x)/(1 - x))^n. - Paul Barry, Jan 20 2008
a(n) is also the number of order-preserving partial transformations (of an n-element chain). Equivalently, it is the order of the semigroup (monoid) of order-preserving partial transformations (of an n-element chain), PO sub n. - Abdullahi Umar, Aug 25 2008
Hankel transform is A180966. - Paul Barry, Sep 29 2010

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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
A. Laradji and A. Umar, Asymptotic results for semigroups of order-preserving partial transformations Comm. Algebra 34 (2006), 1071-1075. [From _Abdullahi Umar_, Oct 11 2008]
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.

Essentially identical to A002003.

Cf. A006318, A122542, A180966.

[1] cat [n le 2 select 2*4^(n-1) else (4*(3*(n-1)^2-1)*Self(n-1) - (2*n-1)*(n-2)*Self(n-2))/((2*n-3)*(n)): n in [1..30]]; // G. C. Greubel, Jul 19 2023

a[n_]:= a[n]= Sum[Binomial[n+k-1,k]*Binomial[n,k], {k,0,n}];
Table[a[n], {n,0,30}]

def A123164(n): return sum(binomial(n,j)*binomial(n+j-1,j) for j in range(n+1))
[A123164(n) for n in range(31)] # G. C. Greubel, Jul 19 2023

a(n) = A122542(2*n,n). - Philippe Deléham, May 28 2007
a(n) = Sum_{k=0..n} C(n, k)*C(n+k-1, k). - Paul Barry, Aug 22 2007
(2*n-1)*(n+1)*a(n+1) = 4*(3*n^2-1)*a(n) - (2*n+1)*(n-1)*a(n-1) for n >= 1 with a(0) = 1 and a(1) = 2. - Abdullahi Umar, Aug 25 2008
a(n) = Jacobi_P(n, 0, -1, 3). - Paul Barry, Sep 27 2009
G.f.: (1 + x + sqrt(1 - 6*x + x^2))/(2*sqrt(1 - 6*x + x^2)). - Paul Barry, Sep 29 2010
From Abdullahi Umar, Oct 11 2008: (Start)
a(n+1) - a(n) = (2*n + 1)*A006318 (n >= 0);
2*a(n) = (n + 1)*A006318(n) - (n - 1)*A006318(n-1) (n > 0). (End)
a(n) = Hypergeometric2F1([-n, n], [1], -1). - Peter Luschny, Aug 02 2014
a(n) ~ (1 + sqrt(2))^(2*n) / (2^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021
From Peter Bala, Oct 07 2021: (Start)
a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*C(n, k)*C(3*n-2*k-1, n-2*k).
a(p) == 2 (mod p^3) for prime p >= 5.
Conjecture: a(n*p^k) == a(n*p^(k-1)) mod( p^(3*k) ) for prime p >= 5 and all positive integers n and k. (End)

Edited by N. J. A. Sloane, Oct 04 2006

Offset changed (a(0)=1) by Michael Somos, Feb 07 2011

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 4, 30, 60, 35, 0, 5, 60, 210, 280, 126, 0, 6, 105, 560, 1260, 1260, 462, 0, 7, 168, 1260, 4200, 6930, 5544, 1716, 0, 8, 252, 2520, 11550, 27720, 36036, 24024, 6435, 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310

Offset: 0

Peter Luschny, Mar 20 2015

The rows give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1) / (n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022

This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - F. Chapoton, Nov 17 2022.

			[1]
[0, 1]
[0, 2,   3]
[0, 3,  12,   10]
[0, 4,  30,   60,   35]
[0, 5,  60,  210,  280,  126]
[0, 6, 105,  560, 1260, 1260,  462]
[0, 7, 168, 1260, 4200, 6930, 5544, 1716]
.
R_0(x) = 1/(x-1)^0.
R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
.
Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
[0] 1, 1,   3,   10,    35,    126,     462, ...
[1] 0, 2,  12,   60,   280,   1260,    5544, ...
[2] 0, 3,  30,  210,  1260,   6930,   36036, ...
[3] 0, 4,  60,  560,  4200,  27720,  168168, ...
[4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
[5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
[6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...

Seiichi Manyama, Rows n = 0..139, flattened
F. Chapoton Enumerative Properties of Generalized Associahedra, Sém. Loth. Comb. B51b (2004).
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and A. Zelevinsky Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3.
Yunlong Shen and Lixin Shen, Orthogonal Fourier-Mellin moments for invariant pattern recognition, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).

T(n, n) = C(2*n-1, n) = A001700(n-1).
T(n, n-1) = A005430(n-1) for n >= 1.
T(n, n-2) = A051133(n-2) for n >= 2.
T(n, 2) = A027480(n-1) for n >= 2.
T(2*n, n) = A208881(n) for n >= 0.
A002002 (row sums).
Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

T_row := proc(n) local egf, k, F, t;
if n=0 then RETURN(1) fi;
egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
F := convert(t,parfrac,x);
# print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
# gives A000142, A001286, A001754, A001755, A001777, ...
seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
seq(print(T_row(n)),n=0..7);
# 2nd version by R. J. Mathar, Dec 18 2016:
A253283 := proc(n,k)
    binomial(n,k)*binomial(n+k-1,k-1) ;
end proc:

Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 29 2018

The exponential generating functions for the rows of the square array L(n,k) = ((n+k)!/n!)*C(n+k-1,n-1) (associated to the unsigned Lah numbers) are given by R_n(x) = Sum_{k=0..n} T(n,k)/(x-1)^(n+k).
T(n,k) = C(n,k)*C(n+k-1,k-1).
Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
Row generating function: Sum_{k>=1} T(n,k)*z^k = z*n* 2F1(1-n,n+1 ; 2; -z). - R. J. Mathar, Dec 18 2016
From Peter Bala, Feb 22 2017: (Start)
G.f.: (1/2)*( 1 + (1 - t)/sqrt(1 - 2*(2*x + 1)*t + t^2) ) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
The row polynomials are the black diamond product of the polynomials x^n and x^(n+1) (see Dukes and White 2016 for the definition of this product).
exp(Sum_{n >= 1} R(n,x)*t^n/n) = 1 + x*t + x*(1 + 2*x)*t^2 + x*(1 + 5*x + 5*x^2)*t^3 + ... is a g.f. for A033282, but with a different offset.
The polynomials P(n,x) := (-1)^n/n!*x^(2*n)*(d/dx)^n(1 + 1/x)^n begin 1, 3 + 2*x , 10 + 12*x + 3*x^2, ... and are the row polynomials for the row reverse of this triangle. (End)
Let Q(n, x) = Sum_{j=0..n} (-1)^(n - j)*A269944(n, j)*x^(2*j - 1) and P(x, y) = (LegendreP(x, 2*y + 1) - LegendreP(x-1, 2*y + 1)) / 2 (see Peter Bala above). Then n!*(n - 1)!*[y^n] P(x, y) = Q(n, x) for n >= 1. - Peter Luschny, Oct 31 2022
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 0} binomial(2*n-1, n)*(x*t)^n/(1 - t)^(2*n) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
It follows that for integer x, the sequence {R(n, x) : n >= 0} satisfies the Gauss congruences: R(n*p^r, x) == R(n*p^(r-1), x) (mod p^r) for all primes p and positive integers n and r.
R(n, -2) = (-1)^n * A002003(n) for n >= 1.
R(n, 3) = A299507(n). (End)

A064861 Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 8, 4, 1, 6, 13, 12, 4, 1, 8, 25, 38, 28, 8, 1, 9, 33, 63, 66, 36, 8, 1, 11, 51, 129, 192, 168, 80, 16, 1, 12, 62, 180, 321, 360, 248, 96, 16, 1, 14, 86, 304, 681, 1002, 968, 592, 208, 32, 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32, 1, 17

Offset: 0

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

When A064861 is regarded as a triangle read by rows, this is [1,0,-1,0,0,0,0,0,0,...] DELTA [2,-1,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 14 2008

			Table begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  2,  3,  5,   6,   8,  9, 11, ...
  2,  8, 13,  25,  33, 51, ...
  4, 12, 38,  63, 129, ...
  4, 28, 66, 192, ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
C. de Jesús Pita Ruiz Velasco, Convolution and Sulanke Numbers, JIS 13 (2010) 10.1.8.
R. A. Sulanke, Problem 10894, Amer. Math. Monthly 108, (2001), p. 770.

Cf. central Delannoy numbers a(n,n) = A001850(n), Delannoy numbers (same main diagonal): a(n,n) = A008288(n,n), a(n-1,n)=A002003(n), a(n,n+1)=A002002(n), a(n,1)=A058582(n), apparently a(n,n+2)=A050151(n).

a064861 n k = a064861_tabl !! n !! k
a064861_row n = a064861_tabl !! n
a064861_tabl = map fst $ iterate f ([1], 2) where
f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 3 - z)
-- Reinhard Zumkeller, May 01 2014

A064861 := proc(n,k) option remember; if n = 1 then 1; elif k = 0 then 0; else procname(n,k-1)+(3/2-1/2*(-1)^(n+k))*procname(n-1,k); fi; end;
seq(seq(A064861(i,j-i),i=1..j-1),j=1..19);

max = 12; se = Series[(1 + 2*x + y*x)/(1 - 2*x^2 - y^2*x^2 - 3*y*x^2), {x, 0, max}, {y, 0, max}]; cc = CoefficientList[se, {x, y}]; Flatten[ Table[ cc[[n, k]], {n, 1, max}, {k, n, 1, -1}]] (* Jean-François Alcover, Oct 21 2011, after g.f. *)

a(n,m)=if(n<0 || m<0,0,polcoeff(polcoeff((1+2*x+y*x)/(1-2*x^2-y^2*x^2-3*y*x^2)+O(x^(n+m+1)),n+m),m))

G.f.: Sum_{m>=0} Sum_{n>=0} a_{m, n}*t^m*s^n = A(t,s) = (1+2*t+s)/(1-2*t^2-s^2-3*s*t).

cp's OEIS Frontend

A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).

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

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A035607 Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).

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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.

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A103885 a(n) = [x^(2*n)] ((1 + x)/(1 - x))^n.

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