A050151 -id:A050151 - cp's OEIS frontend

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

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

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).

A118384 Gaussian column reduction of Hankel matrix for central Delannoy numbers.

Original entry on oeis.org

1, 3, 1, 13, 6, 1, 63, 33, 9, 1, 321, 180, 62, 12, 1, 1683, 985, 390, 100, 15, 1, 8989, 5418, 2355, 720, 147, 18, 1, 48639, 29953, 13923, 4809, 1197, 203, 21, 1, 265729, 166344, 81340, 30744, 8806, 1848, 268, 24, 1, 1462563, 927441, 471852, 191184, 60858

Offset: 0

Paul Barry, Apr 26 2006

First column is central Delannoy numbers A001850. Second column is A050151.

			Triangle begins:
     1,
     3,     1,
    13,     6,     1,
    63,    33,     9,     1,
   321,   180,    62,    12,    1,
  1683,   985,   390,   100,   15,   1

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 19.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
W.-J. Woan, Hankel Matrices and Lattice Paths, J. Integer Sequences, 4 (2001), #01.1.2.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Cf. A110171, A376467.

Table[Sum[Binomial[n,i]Binomial[n,n-k-i]2^i,{i,0,n-k}],{n,0,8},{k,0,8}]//MatrixForm

create_list(sum(binomial(n,i)*binomial(n,n-k-i)*2^i,i,0,n),n,0,8,k,0,n);

Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*C(j,n-k-j)*2^(n-k-j)*3^(2*j-(n-k));
Riordan array (1/sqrt(1-6*x+x^2), (1-3*x-sqrt(1-6*x+x^2))/(4*x));
Column k has e.g.f. exp(3*x)*Bessel_I(k,2*sqrt(2)x)/(sqrt(2))^k.
a(n,k) = Sum_{i = 0..n} binomial(n,i)*binomial(n,n-k-i)*2^i, also a(n+1,k+1) = a(n,k) + 3*a(n,k+1) + 2*a(n,k+2). - Emanuele Munarini, Mar 16 2011
From Peter Bala, Jun 29 2015: (Start)
Matrix product A110171 * A007318.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - 3*x - sqrt(1 - 6*x + x^2) )/(4*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan, Jan 2000, Example 5.2).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + 3*x + 2*x^2. In general the (n,k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2kx + ((k-2)x)^2 + (1 - kx) * sqrt(1 - 2kx + ((k-2)*x)^2)).

Original entry on oeis.org

1, 1, 0, 1, 2, -3, 1, 4, 3, 0, 1, 6, 15, 4, 10, 1, 8, 33, 56, 5, 0, 1, 10, 57, 180, 210, 6, -35, 1, 12, 87, 400, 985, 792, 7, 0, 1, 14, 123, 740, 2810, 5418, 3003, 8, 126, 1, 16, 165, 1224, 6285, 19824, 29953, 11440, 9, 0, 1, 18, 213, 1876, 12130, 53550, 140497, 166344, 43758, 10, -462

Offset: 0

Seiichi Manyama, Jan 26 2020

			Square array begins:
   1, 1,   1,    1,     1,     1, ...
   0, 2,   4,    6,     8,    10, ...
  -3, 3,  15,   33,    57,    87, ...
   0, 4,  56,  180,   400,   740, ...
  10, 5, 210,  985,  2810,  6285, ...
   0, 6, 792, 5418, 19824, 53550, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.
T(n,n+1) gives A331794.
Cf. A307883, A331511, A331514, A331795.

T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)

T(n,k) = Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (k-1)^j * k^(n-2*j) * binomial(n+1,n-2*j) * binomial(2*j+1,j). - Seiichi Manyama, Aug 24 2025
From Seiichi Manyama, Aug 27 2025: (Start)
T(n,k) = [x^n] (1+k*x+(k-1)*x^2)^(n+1).
For k != 1, e.g.f. of column k: exp(k*x) * BesselI(1, 2*sqrt(k-1)*x) / sqrt(k-1), with offset 1. (End)

A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).

Original entry on oeis.org

1, 2, 1, 8, 4, 1, 38, 18, 6, 1, 192, 88, 32, 8, 1, 1002, 450, 170, 50, 10, 1, 5336, 2364, 912, 292, 72, 12, 1, 28814, 12642, 4942, 1666, 462, 98, 14, 1, 157184, 68464, 27008, 9424, 2816, 688, 128, 16, 1, 864146, 374274, 148626, 53154, 16722, 4482, 978, 162, 18, 1

Offset: 0

Emeric Deutsch, Jul 14 2005

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Column k for k >= 1 has g.f. z^k*R^(k-1)*g*(1+z*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1-6z+z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318) and g = 1/sqrt(1-6z+z^2) is the g.f. of the central Delannoy numbers (A001850).
Sum_{k=0..n} k*T(n,k) = A050151(n) (the partial sums of the central Delannoy numbers) = (1/2)*n*R(n), where R(n) = A006318(n) is the n-th large Schroeder number.
From Paul Barry, May 07 2009: (Start)
Riordan array ((1+x+sqrt(1-6x+x^2))/(2*sqrt(1-6x+x^2)), (1-x-sqrt(1-6x+x^2))/2).
Inverse of Riordan array ((1-2x-x^2)/(1-x^2), x(1-x)/(1+x)). (End)

			T(2,1)=4 because we have NED, NENE, NEEN and NDE.
Triangle starts:
    1;
    2,  1;
    8,  4,  1;
   38, 18,  6,  1;
  192, 88, 32,  8,  1;
From _Paul Barry_, May 07 2009: (Start)
Production matrix is
   2, 1,
   4, 2, 1,
   6, 2, 2, 1,
   8, 2, 2, 2, 1,
  10, 2, 2, 2, 2, 1,
  12, 2, 2, 2, 2, 2, 1,
  14, 2, 2, 2, 2, 2, 2, 1,
  16, 2, 2, 2, 2, 2, 2, 2, 1,
  18, 2, 2, 2, 2, 2, 2, 2, 2, 1 (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
W.-j. Woan, The Lagrange Inversion Formula and Divisibility Properties, JIS 10 (2007) 07.7.8, example 5.

Cf. A001850, A002003, A006318, A050146, A050151, A113139, A118384.

Q:=sqrt(1-6*z+z^2): G:=(1+z+Q)/Q/(2-t+t*z+t*Q): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

T[n_, n_] = 1;
T[n_, k_] := Sum[Binomial[n, i] Binomial[2n-k-i-1, n-k-i], {i, 0, n}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 13 2019 *)

A110171 = lambda n,k : binomial(n, k)*hypergeometric([k-n, n], [k+1], -1)
for n in (0..9): [round(A110171(n,k).n(100)) for k in (0..n)] # Peter Luschny, Sep 17 2014

T(n,0) = A002003(n) for n >= 1.
T(n,1) = A050146(n) for n >= 1.
Row sums are the central Delannoy numbers (A001850).
G.f.: (1+z+Q)/(Q(2-t+tz+tQ)), where Q=sqrt(1-6z+z^2).
T(n,k) = x^(n-k)*((1+x)/(1-x))^n. - Paul Barry, May 07 2009
T(n,k) = C(n, k)*hypergeometric([k-n, n], [k+1], -1). - Peter Luschny, Sep 17 2014
From Peter Bala, Jun 29 2015: (Start)
T(n,k) = Sum_{i = 0..n} binomial(n,i)*binomial(2*n-k-i-1,n-k-i).
Matrix product A118384 * A007318^(-1)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - x - sqrt(1 - 6*x + x^2) )/2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan). (End)
T(n,k) = P(n-k, k, -1, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta. Cf. A113139. - Peter Bala, Feb 16 2020

A387337 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k) * binomial(n+2,k+2).

Original entry on oeis.org

1, 9, 62, 390, 2355, 13923, 81340, 471852, 2725365, 15698925, 90278298, 518620050, 2977523367, 17089401735, 98073111800, 562837648728, 3230477449641, 18545060127249, 106484493098230, 611580138259230, 3513476242850651, 20190255588746859, 116056618927038132

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A050151.

[&+[2^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Table[Sum[2^k*Binomial[n+2, k] * Binomial[n+2, k+2], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Aug 27 2025 *)

a(n) = sum(k=0, n, 2^k*binomial(n+2, k)*binomial(n+2, k+2));

n*(n+4)*a(n) = (n+2) * (3*(2*n+3)*a(n-1) - (n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+3*x+2*x^2)^(n+2).
E.g.f.: exp(3*x) * BesselI(2, 2*sqrt(2)*x) / 2, with offset 2.
a(n) ~ (1 + sqrt(2))^(2*n+5) / (2^(9/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 27 2025

A277060 a(n) = (1/2) * Sum_{k=0..n} (binomial(n,k) * binomial(n+k,k+1))^2 for n >= 0.

Original entry on oeis.org

0, 1, 28, 729, 19376, 529575, 14835780, 424231465, 12338211520, 363931754949, 10862528888300, 327501958094003, 9959845931792784, 305175084350065267, 9412306255856822388, 291982561878565118025, 9104382992541189221120

Offset: 0

Seiichi Manyama, Nov 07 2016

Conjecture: the supercongruences a(p-1) == 1 (mod p^4) holds for all primes p >= 5 and a(p^2-1) == 1 (mod p^5) holds for all primes p >= 3. - Peter Bala, Mar 22 2023

Seiichi Manyama, Table of n, a(n) for n = 0..656

Cf. 1/2 * Sum_{k=0..n} (binomial(n,k) * binomial(n+k,k+1))^m: A050151 (m=1), this sequence (m=2).

Cf. A005259, A074635.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else ( (2*n-1)*(51*n^4-102*n^3+19*n^2+ 32*n-14)*n^2*a(n-1) - n^2*(n-2)*(3*n^2-1)*(n-1)^2*a(n-2) )/( (n-1)^2*(3*n^2-6*n+2)*(n+1)^3 ) end if; end:
seq(a(n), n = 0..20); # Peter Bala, Mar 22 2023

a(n)=my(t=n); if(n<2, return(n)); sum(k=1,n, t*=(n-k+1)*(n+k)/k/(k+1); t^2, n^2)/2 \\ Charles R Greathouse IV, Nov 07 2016

a(n) = n^2 * A074635(n)/2.
From Peter Bala, Mar 22 2023: (Start)
a(n) = Sum_{k = 0..n-1} binomial(n+1,k)*binomial(n-1,k)*binomial(n+k,k)^2.
P-recursive: (n-1)^2*(3*n^2-6*n+2)*(n+1)^3*a(n) = (2*n-1)*(51*n^4-102*n^3+19*n^2+ 32*n-14)*n^2*a(n-1) - n^2*(n-2)*(3*n^2-1)*(n-1)^2*a(n-2) with a(0) = 0 and a(1) = 1.
a(n) ~ sqrt(12 + 17*sqrt(2)/2)*(17 + 12*sqrt(2))^n/(4*n^(3/2)*Pi^(3/2)). (End)

A272865 Triangle read by rows, T(n,k) are covariances of inverse power traces of complex Wishart matrices with parameter c=2, for n>=1 and 1<=k<=n.

Original entry on oeis.org

4, 24, 160, 132, 936, 5700, 720, 5312, 33264, 198144, 3940, 29880, 190980, 1155600, 6823620, 21672, 167712, 1088856, 6670656, 39786120, 233908896, 119812, 941640, 6189540, 38300976, 230340740, 1363667256, 7997325700

Offset: 1

Fabio Deelan Cunden, May 08 2016

These numbers provide the covariances of power traces of the time-delay matrix when the scattering matrix belongs to the Dyson ensembles.

Relation with A047781 and A002002. See eq. (60) and (61) in Cunden et al., J. Phys. A: Math. Theor. 49, 18LT01 (2016).

			Triangle starts:
4;
24,   160;
132,  936,   5700;
720,  5312,  33264,  198144;
3940, 29880, 190980, 1155600, 6823620;

F. D. Cunden, "Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities", Phys. Rev. E 91, 060102(R) (2015).
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, "Correlators for the Wigner-Smith time-delay matrix of chaotic cavities", J. Phys. A: Math. Theor. 49, 18LT01 (2016).
F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, "Moments of Random Matrices and Hypergeometric Orthogonal Polynomials", Commun. Math. Phys. 369, 1091-1145 (2019).

F. D. Cunden, Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities, arXiv:1412.2172 [cond-mat.mes-hall], 2014-2015.
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.
F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, Moments of Random Matrices and Hypergeometric Orthogonal Polynomials, arXiv:1805.08760 [math-ph], 2018.

Cf. A002002, A047781, A050151.

P := (n,k) -> simplify(n*hypergeom([1-k,k+1],[1],-1)*hypergeom([1-n,n+1],[2],-1)): seq(seq(4*(n*k)*(P(n,k)+P(k,n))/(n+k),k=1..n),n=1..7); # Peter Luschny, May 08 2016

Clear["Global`*"];(*Wigner-Smith Covariance*)
P[k_] := Sum[Binomial[k - 1, j] Binomial[k + j, j], {j, 0, k - 1}]
Q[k_] := Sum[Binomial[k, j + 1] Binomial[k + j, j], {j, 0, k - 1}]
a[k1_, k2_] := 4 (k1 k2)/(k1 + k2) (P[k1] Q[k2] + P[k2] Q[k1])
L = 10; Table[a[k, l], {k, 1, L}, {l, 1, k}]

G.f.: ((x*y)/(x-y)^2)*((x*y-3(x+y)+1)/(sqrt(x^2-6x+1)*sqrt(y^2-6y+1))-1).

T(n,1)/4 = A050151(n) for n>=1. - Peter Luschny, May 08 2016

A350519 a(n) = A(n,n) where A(1,n) = A(n,1) = prime(n+1) and A(m,n) = A(m-1,n) + A(m,n-1) + A(m-1,n-1) for m > 1 and n > 1.

Original entry on oeis.org

3, 13, 63, 325, 1719, 9237, 50199, 275149, 1518263, 8422961, 46935819, 262512929, 1472854451, 8285893713, 46723439019, 264009961733, 1494486641911, 8473508472009, 48112827862527, 273541139290857, 1557023508876891, 8872219429659729, 50605041681538595, 288897992799897481

Offset: 1

Yigit Oktar, Jan 02 2022

nonn

Replacing prime(n+1) by other functions f(n) we can get many other sequences. For example, with f(n) = 1 we get A001850.

			The two-dimensional recurrence A(m,n) can be depicted in matrix form as
   3   5   7   11   13    17    19 ...
   5  13  25   43   67    97   133 ...
   7  25  63  131  241   405   635 ...
  11  43 131  325  697  1343  2383 ...
  13  67 241  697 1719  3759  7485 ...
  17  97 405 1343 3759  9237 20481 ...
  19 133 635 2383 7485 20481 50199 ...
  ...
and then a(n) is the main diagonal of this matrix, A(n,n).

Cf. A000040, A001850, A002002, A050151, A344576 (see comments).

clear all
close all
sz = 14
f = zeros(sz,sz);
pp = primes(50);
f(1,:) = pp(2:end);
f(:,1) = pp(2:end);
for m=2:sz
    for  n=2:sz
        f(m,n) = f(m-1,n-1)+f(m,n-1)+f(m-1,n);
    end
end
an = []
for n=1:sz
    an = [an f(n,n)];
end
S = sprintf('%i,',an);
S = S(1:end-1)

f[1,1]=3;f[m_,1]:=Prime[m+1];f[1,n_]:=Prime[n+1];f[m_,n_]:=f[m,n]=f[m-1,n]+f[m,n-1]+f[m-1,n-1];Table[f[n,n],{n,25}] (* Giorgos Kalogeropoulos, Jan 03 2022 *)

cp's OEIS Frontend

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

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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.

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A118384 Gaussian column reduction of Hankel matrix for central Delannoy numbers.

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A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).

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A110171 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).

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A387337 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k) * binomial(n+2,k+2).

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A277060 a(n) = (1/2) * Sum_{k=0..n} (binomial(n,k) * binomial(n+k,k+1))^2 for n >= 0.

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A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2kx + ((k-2)x)^2 + (1 - kx) * sqrt(1 - 2kx + ((k-2)*x)^2)).