A063007 -id:A063007 - cp's OEIS frontend

A115951 Expansion of 1/sqrt(1-4xy-4x^2y).

1, 0, 2, 0, 2, 6, 0, 0, 12, 20, 0, 0, 6, 60, 70, 0, 0, 0, 60, 280, 252, 0, 0, 0, 20, 420, 1260, 924, 0, 0, 0, 0, 280, 2520, 5544, 3432, 0, 0, 0, 0, 70, 2520, 13860, 24024, 12870, 0, 0, 0, 0, 0, 1260, 18480, 72072, 102960, 48620, 0, 0, 0, 0, 0, 252, 13860, 120120, 360360, 437580, 184756

Offset: 0

Paul Barry, Mar 14 2006

Row sums are A006139. Diagonal sums are A115962.

Coefficients of 2^n * P(n, x) with P the Legendre P polynomials. Reflection of triangle A008556. - Ralf Stephan, Apr 07 2016.

			Triangle begins
   1,
   0,  2,
   0,  2,  6,
   0,  0, 12,  20,
   0,  0,  6,  60,  70,
   0,  0,  0,  60, 280,  252,
   0,  0,  0,  20, 420, 1260, 924

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A006139 (row sums), A063007 (binomial transform), A115962 (diagonal sums).

/* As triangle */ [[Binomial(2*k,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

Table[Binomial[2k, k]Binomial[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Sep 02 2015 *)

{T(n,k) = binomial(2*k,k)*binomial(k,n-k)}; \\ G. C. Greubel, May 06 2019

[[binomial(2*k,k)*binomial(k,n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 06 2019

Number triangle T(n,k) = C(2k,k)*C(k,n-k).
From Peter Bala, Sep 02 2015: (Start)
Binomial transform is A063007; equivalently, P * M = A063007, where P denotes Pascal's triangle A007318 and M denotes the present array.
P * M * P^-1 is a signed version of A063007. (End)

A080721 Triangle of binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1)).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 21, 14, 1, 16, 66, 100, 50, 1, 25, 160, 410, 455, 182, 1, 36, 330, 1260, 2310, 2016, 672, 1, 49, 609, 3220, 8610, 12222, 8778, 2508, 1, 64, 1036, 7224, 26250, 53592, 61908, 37752, 9438, 1, 81, 1656, 14700, 69300, 189882, 312312, 303732, 160875, 35750, 1

Offset: 0

Paul Boddington, Mar 07 2003

For n>1 and 0 <= k <= n, a(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's 'cluster algebra' of finite type D_n.

Triangle of f-vectors of the simplicial complexes dual to the generalized associahedra of type D_n (n >= 2). See A145903 for the corresponding triangle of h-vectors. For the triangles of f-vectors of type A and type B associahedra see A033282 and A063007 respectively. [Peter Bala, Oct 28 2008]

			Contribution from _Peter Bala_, Oct 28 2008: (Start)
Triangle begins
n\k|..0....1....2....3....4....5
================================
0..|..1
1..|..1....1
2..|..1....4....4
3..|..1....9...21...14
4..|..1...16...66..100...50
5..|..1...25..160..410..455..182
...
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001; J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
Yasuaki Gyoda, Positive cluster complexes and tau-tilting simplicial complexes of cluster-tilted algebras of finite type, arXiv:2105.07974 [math.RT], 2021, see page 34.

Cf. A033282, A063007.

A051924 (main diagonal), A145903( h-vectors type D associahedra). [From Peter Bala, Oct 28 2008]

A080721 := proc(n,k)
    binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
end proc: # R. J. Mathar, Mar 22 2013

Flatten[Table[Binomial[n,k](Binomial[n+k,k]-Binomial[Abs[n+k-2],k-1]),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Feb 20 2013 *)

T(n,k)=binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1))
for (n=0, 10, for (k=0,n, print1(T(n,k),", ")));
/* Joerg Arndt, Feb 21 2013 */

A178301 Triangle T(n,k) = binomial(n,k)*binomial(n+k+1,n+1) read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 8, 10, 1, 15, 45, 35, 1, 24, 126, 224, 126, 1, 35, 280, 840, 1050, 462, 1, 48, 540, 2400, 4950, 4752, 1716, 1, 63, 945, 5775, 17325, 27027, 21021, 6435, 1, 80, 1540, 12320, 50050, 112112, 140140, 91520, 24310, 1, 99, 2376, 24024, 126126, 378378, 672672, 700128, 393822, 92378

Offset: 0

Alford Arnold, May 30 2010

Antidiagonal sums are given by A113682. - Johannes W. Meijer, Mar 24 2013
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial binomial(x+n,n)*binomial(x+n,n-1) in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
Chapoton's observation above is correct: the precise expansion is  binomial(x+n,n)*binomial(x+n,n-1) = Sum_{k = 0..n-1} (-1)^k*T(n-1,n-1-k)*binomial(x+2*n-1-k,2*n-1-k), as can be verified using the WZ algorithm. For example, n = 4 gives binomial(x+4,4)*binomial(x+4,3)  = 35*binomial(x+7,7) - 45*binomial(x+6,6) + 15*binomial(x+5,5) - binomial(x+4,4). - Peter Bala, Jun 24 2023

			n=0: 1;
n=1: 1,  3;
n=2: 1,  8,  10;
n=3: 1, 15,  45,   35;
n=4: 1, 24, 126,  224,   126;
n=5: 1, 35, 280,  840,  1050,   462;
n=6: 1, 48, 540, 2400,  4950,  4752,  1716;
n=7: 1, 63, 945, 5775, 17325, 27027, 21021, 6435;

David A. Corneth, Table of n, a(n) for n = 0..10010
Author?, Norm of a continuous function, dxdy.ru (in Russian).

Cf. A007318, A047781 (row sums), A178300, A182626, A123160, A132813.

A178301 := proc(n,k)
        binomial(n,k)*binomial(n+k+1,n+1) ;
end proc: # R. J. Mathar, Mar 24 2013
R := proc(n) add((-1)^(n+k)*(2*k+1)*orthopoly:-P(k,2*x+1)/(n+1), k=0..n) end:
for n from 0 to 6 do seq(coeff(R(n), x, k), k=0..n) od; # Peter Luschny, Aug 25 2021

Flatten[Table[Binomial[n,k]Binomial[n+k+1,n+1],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Aug 23 2014 *)

create_list(binomial(n,k)*binomial(n+k+1,n+1),n,0,12,k,0,n); /* Emanuele Munarini, Dec 16 2016 */

R(n,x) = sum(k=0,n, (-1)^(n+k) * (2*k+1) * pollegendre(k,2*x+1)) / (n+1); \\ Max Alekseyev, Aug 25 2021

T(n,k) = A007318(n,k) * A178300(n+1,k+1).
From Peter Bala, Jun 18 2015: (Start)
n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k+1,n+1)*x^k = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+1,k+1)*binomial(n+k+1,n+1)*(1 + x)^k.
Recurrence: (2*n - 1)*(n + 1)*R(n,x) = 2*(4*n^2*x + 2*n^2 - x - 1)*R(n-1,x) - (2*n + 1)(n - 1)*R(n-2,x) with R(0,x) = 1, R(1,x) = 1 + 3*x.
A182626(n) = -R(n-1,-2) for n >= 1. (End)
From Peter Bala, Jul 20 2015: (Start)
n-th row polynomial R(n,x) = Jacobi_P(n,0,1,2*x + 1).
(1 + x)*R(n,x) gives the row polynomials of A123160. (End)
G.f.: (1+x-sqrt(1-2*x+x^2-4*x*y))/(2*(1+y)*x*sqrt(1-2*x+x^2-4*x*y)). - Emanuele Munarini, Dec 16 2016
R(n,x) = Sum_{k=0..n} (-1)^(n+k)*(2*k+1)*P(k,2*x+1)/(n+1), where P(k,x) is the k-th Legendre polynomial (cf. A100258) and P(k,2*x+1) is the k-th shifted Legendre polynomial (cf. A063007). - Max Alekseyev, Jun 28 2018; corrected by Peter Bala, Aug 08 2021
Polynomial g(n,x) = R(n,-x)/(n+1) delivers the maximum of f(1)^2/(Integral_{x=0..1} f(x)^2 dx) over all polynomials f(x) with real coefficients and deg(f(x)) <= n. This maximum equals (n+1)^2. See dxdy.ru link. - Max Alekseyev, Jun 28 2018

A110098 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n, having k return steps to the line y = x from the line y = x+1 (i.e., E steps from the line y=x+1 to the line y = x).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 22, 30, 10, 1, 90, 146, 70, 14, 1, 394, 714, 430, 126, 18, 1, 1806, 3534, 2490, 938, 198, 22, 1, 8558, 17718, 14002, 6314, 1734, 286, 26, 1, 41586, 89898, 77550, 40054, 13338, 2882, 390, 30, 1, 206098, 461010, 426150, 244790, 94554, 24970, 4446, 510, 34, 1

Offset: 0

Emeric Deutsch, Jul 11 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).
The row sums are the central Delannoy numbers (A001850).
Column 0 yields the large Schroeder numbers (A006318).
Column 1 yields A006320.
Column k has g.f. z^k*R^(2*k+1), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).

			T(2, 1) = 6 because we have DN(E), N(E)D, N(E)EN, ND(E), NNE(E) and ENN(E) (the return E steps are shown between parentheses).
Triangle begins:
   1;
   2,   1;
   6,   6,   1;
  22,  30,  10,   1;
  90, 146,  70,  14,   1;

T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, example p. 37.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A063007, A006318, A006320, A033877, A110099, A110107.

T := proc(n, k) if k=n then 1 else ((2*k+1)/(n-k))*sum(binomial(n-k,j)*binomial(n+k+j,n-k-1),j=0..n-k) fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

T[n_, k_] := If[k == n, 1, ((2*k+1)/(n-k))*Sum[Binomial[n-k, j]*Binomial[n+k+j, n-k-1], {j, 0, n-k}]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 21 2024, after Maple program *)

T(n,k) = ((2*k+1)/(n-k))*Sum_{j=0..n-k} binomial(n-k, j)*binomial(n+k+j, n-k-1) for k < n;
T(n,n) = 1;
T(n,k) = 0 for k > n.
G.f.: R/(1 - t*z*R^2), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).
Sum_{k=0..n} k*T(n,k) = A110099(n).
T(n,k) = A033877(n-k+1, n+k+1). - Johannes W. Meijer, Sep 05 2013
It appears that this triangle equals M * N^(-1), where M is the lower triangular array A063007 and N = ( (-1)^(n+k)* binomial(n, k)*binomial(n+k, k) )n,k >= 0 is a signed version of A063007. - Peter Bala, Oct 07 2024

A190909 Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 6, 1, 15, 70, 168, 54, 30, 1, 21, 140, 504, 270, 330, 20, 1, 28, 252, 1260, 990, 1980, 260, 140, 1, 36, 420, 2772, 2970, 8580, 1820, 2100, 70, 1, 45, 660, 5544, 7722, 30030, 9100, 16800, 1190, 630

Offset: 0

Peter Luschny, May 24 2011

The triangle may be regarded as a generalization of the triangle A063007.
A063007(n,k) = binomial(n+k, n-k)*(2*k)$;
T(n,k) = binomial(n+k, n-k)*(k)$.
Here n$ denotes the swinging factorial A056040(n). As A063007 is a decomposition of the central Delannoy numbers A001850, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
T(n,n) = A056040(n) which can be seen as extended central binomial numbers.

			[0]  1
[1]  1,  1
[2]  1,  3,   2
[3]  1,  6,  10,    6
[4]  1, 10,  30,   42,   6
[5]  1, 15,  70,  168,  54,   30
[6]  1, 21, 140,  504, 270,  330,  20
[7]  1, 28, 252, 1260, 990, 1980, 260, 140

Peter Luschny, The lost Catalan numbers
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Cf. Row sums: A190910; A056040, A063007, A085478, A088617.

A190909 := (n,k) -> binomial(n+k,n-k)*k!/iquo(k,2)!^2:
seq(print(seq(A190909(n,k),k=0..n)),n=0..7);

Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 25 2012 *)

T(n,1) = A000217(n). T(n,2) = 2*binomial(n+2,4) (Cf. A034827).

A156763 Triangle T(n, k) = binomial(2k, k)binomial(n+k, n-k) + binomial(2n-k, k)binomial(2*(n-k), n-k), read by rows.

Original entry on oeis.org

2, 3, 3, 7, 12, 7, 21, 42, 42, 21, 71, 160, 180, 160, 71, 253, 660, 770, 770, 660, 253, 925, 2814, 3570, 3360, 3570, 2814, 925, 3433, 12068, 17388, 15750, 15750, 17388, 12068, 3433, 12871, 51552, 85344, 81312, 69300, 81312, 85344, 51552, 12871

Offset: 0

Roger L. Bagula, Feb 15 2009

			Triangle begins as:
      2;
      3,      3;
      7,     12,      7;
     21,     42,     42,     21;
     71,    160,    180,    160,     71;
    253,    660,    770,    770,    660,    253;
    925,   2814,   3570,   3360,   3570,   2814,    925;
   3433,  12068,  17388,  15750,  15750,  17388,  12068,   3433;
  12871,  51552,  85344,  81312,  69300,  81312,  85344,  51552,  12871;
  48621, 218880, 413820, 438900, 342342, 342342, 438900, 413820, 218880, 48621;

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 66.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001850, A063007.

A063007:= func< n,k | Binomial(n, k)*Binomial(n+k, k) >;
A156763:= func< n,k | A063007(n,k) + A063007(n,n-k) >;
[A156763(n,k): k in [0..n]. n in [0..12]]; // G. C. Greubel, Jun 15 2021

T[n_, k_]:= Binomial[n+k, n-k]*Binomial[2*k, k] + Binomial[2*(n-k), n-k]*Binomial[ 2*n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 15 2021 *)

def A063007(n, k): return binomial(n+k, n-k)*binomial(2*k, k)
def A156763(n, k): return A063007(n,k) + A063007(n,n-k)
flatten([[A156763(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k).
T(n, k) = A063007(n, k) + A063007(n, n-k).
Sum_{k=0..n} T(n, k) = 2*A001850(n). - G. C. Greubel, Jun 15 2021

Edited by G. C. Greubel, Jun 15 2021

A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.

Original entry on oeis.org

0, 1, 5, 9, 29, 115, 251, 65, 5191, 1039, 2035, 10391, 2077, 72703, 58157, 256103, 259703, 1817471, 1817521, 7270009, 1454021, 28567, 67323, 25243, 389467, 21810107, 47982293, 6854599, 9822481, 9895981, 11132213, 66793523, 11755653433, 2351131157, 30564700141, 30564710941, 78708473, 237497419, 237487619, 23511313481, 23511309071, 61129406407, 5557218637, 61129406447, 244517610353

Offset: 0

Wolfdieter Lang, May 16 2018

The corresponding denominators are given in A303989.
The numerators of the rational triangle c_{n,k} are denoted by T(n,k). The triangle c_{n,k} is used to compute Apéry's sequence of rationals a_n = A059415(n)/A059416(n), satisfying a certain three term recurrence, as a(n) = Sum_{k=0..n} c_{n,k}*(binomial(n+k,k)*binomial(n,k))^2 = Sum_{k=0..n} (T(n,k)/A303989(n,k))*A303987(n,k).
The column k = 0 gives the numerators of Zeta3(n) = A007408(n)/A007409(n), with Zeta3(0) := 0.

			The triangle T(n, k) begins:
  n/k      0       1        2        3           4          5           6
  0:       0
  1:       1       5
  2:       9      29      115
  3:     251      65     5191     1039
  4:    2035   10391     2077    72703       58157
  5:  256103  259703  1817471  1817521     7270009    1454021
  6:   28567   67323    25243   389467    21810107   47982293     6854599
  ...
  row n = 7: 9822481 9895981 11132213 66793523 11755653433 2351131157 30564700141 30564710941,
  row n = 8: 78708473 237497419 237487619 23511313481 23511309071 61129406407 5557218637 61129406447 244517610353,
  row n = 9: 19148110939 19237016539 211601625329 211601801729 2750823224027 42320357851 550164649543 550164651163 37411196140169 37411196579209,
  ...
------------------------------------------------------------------------------
The rational triangle c_{n,k} starts:
  n\k        0            1              2                3               4
  0:        0/1
  1:        1/1          5/4
  2:        9/8         29/24         115/96
  3:      251/216       65/54        5191/4320        1039/864
  4:    2035/1728    10391/8640      2077/1728      72703/60480      58157/48384
  ...
  row n = 5:  256103/216000 259703/216000 1817471/1512000 1817521/1512000 7270009/6048000 1454021/1209600,
  ...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.

A. van der Poorten, A proof that Euler missed ..., Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/79), no. 4, 195-203, c_{n,k} in section 4.
Wikipedia, Apery's theorem

Cf. A005259, A007408/A007409, A059415, A059416, A063007, A303987, A303989.

T(n,k) = numerator(sum(m=1, n, 1/m^3) + sum(m=1, k, (-1)^(m-1)/(2*m^3*binomial(n,m)*binomial(n+m,m)))) \\ Jason Yuen, May 27 2025

T(n,k) = numerator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k} (-1)^(m-1)/(2*m^3*B(n,m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n,m) = A063007(n,m).

A303989 Triangle read by rows: denominators of c_{n,k}, n >= 0, k = 0..n, used in the proof that Zeta(3) is irrational.

Original entry on oeis.org

1, 1, 4, 8, 24, 96, 216, 54, 4320, 864, 1728, 8640, 1728, 60480, 48384, 216000, 216000, 1512000, 1512000, 6048000, 1209600, 24000, 56000, 21000, 324000, 18144000, 39916800, 5702400, 8232000, 8232000, 9261000, 55566000, 9779616000, 1955923200, 25427001600, 25427001600, 65856000, 197568000, 197568000, 19559232000, 19559232000, 50854003200, 4623091200, 50854003200, 203416012800

Offset: 0

Wolfdieter Lang, May 16 2018

See A303988 for details, references and links.

			The triangle T(n, k) begins:
  n\k       0       1       2        3         4          5          6
  0:        1
  1:        1       4
  2:        8      24      96
  3:      216      54    4320      864
  4:     1728    8640    1728    60480      48384
  5:   216000  216000 1512000  1512000    6048000    1209600
  6:    24000   56000   21000   324000   18144000   39916800     5702400
  ...
  row n = 7: 8232000 8232000 9261000 55566000 9779616000 1955923200 25427001600 25427001600,
  row n = 8: 65856000 197568000 197568000 19559232000 19559232000 50854003200 4623091200 50854003200 203416012800,
  row n = 9: 16003008000 16003008000 176033088000 176033088000 2288430144000 35206617600 457686028800 457686028800 31122649958400 31122649958400,
  ...
For the first rationals c_{n,k} see A303988.

Cf. A007408/A007409, A063007, A303988.

T(n,k) = denominator(sum(m=1, n, 1/m^3) + sum(m=1, k, (-1)^(m-1)/(2*m^3*binomial(n,m)*binomial(n+m,m)))) \\ Jason Yuen, May 28 2025

T(n, k) = denominator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k} (-1)^(m-1)/(2*m^3*B(n, m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n, m) = A063007(n, m).

A345013 Triangle read by rows, related to clusters of type D.

Original entry on oeis.org

1, 4, 3, 15, 20, 6, 56, 105, 60, 10, 210, 504, 420, 140, 15, 792, 2310, 2520, 1260, 280, 21, 3003, 10296, 13860, 9240, 3150, 504, 28, 11440, 45045, 72072, 60060, 27720, 6930, 840, 36

Offset: 1

F. Chapoton, Sep 30 2021

Let C_{n+1} be the cyclic quiver with n+1 vertices. Empirically, the n-th row is related to the green-mutation partial order on clusters for this quiver, restricted to clusters that do not meet the initial seed.
Apparently, value of the associated polynomials at -2 is A089849, up to sign.
By evaluating the associated polynomials at x-1, one apparently gets A062196.
The rows seem to 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)*(x+n+2) / (n! * (n+2)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022
Chapoton's observation above is correct: the precise expansion is (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*(x+n+2) / (n! * (n+2)!) = Sum_{k = 0..n} (-1)^k*T(n+1,k)*binomial(x+2*n+2-k, 2*n+2-k), as can be verified using the WZ algorithm. For example, n = 2 gives (x+1)^2*(x+2)^2*(x+3)*(x+4)/(2!*4!) = 15*binomial(x+6,6) - 20*binomial(x+5,5) + 6*binomial(x+4,4). - Peter Bala, Jun 24 2023

			Triangle begins:
[1] 1
[2] 4,    3
[3] 15,   20,    6
[4] 56,   105,   60,    10
[5] 210,  504,   420,   140,  15
[6] 792,  2310,  2520,  1260, 280,  21
[7] 3003, 10296, 13860, 9240, 3150, 504, 28
...

Cf. A001791 (T(n,1)), A000217 (T(n,n)), A026002 (row sums), A000012 (alternating row sum), A051924 (number of clusters of type D_n).

Cf. A089849, A062196, A063007, A253283.

row(n) = vector(n, k, k--; (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n); \\ Michel Marcus, Sep 30 2021

def T_row(n):
    return [(n-k)*binomial(n,k)*binomial(2*n-k,n-1)//n for k in range(n)]
for n in range(1, 8): print(T_row(n))

T(n, k) = (n-k)*binomial(n,k)*binomial(2*n-k, n-1)/n, for n >= 1 and 0 <= k < n.
From Peter Bala, Jun 24 2023: (Start)
As conjectured above by Chapoton we have
Sum_{k = 0..n-1} T(n,k)*(x - 1)^k = Sum_{k = 0..n-1} A062196(n-1,k)*x^k and
Sum_{k = 0..n-1} T(n,k)*(-2)^k = (-1)^floor(n/2)*A089849(n) for n >= 1 (both easily verified using the WZ algorithm). (End)

A092370 Triangle read by rows: T(n,k)=(1/2)C(n+k,k)C(n,n-k).

Original entry on oeis.org

1, 3, 3, 6, 15, 10, 10, 45, 70, 35, 15, 105, 280, 315, 126, 21, 210, 840, 1575, 1386, 462, 28, 378, 2100, 5775, 8316, 6006, 1716, 36, 630, 4620, 17325, 36036, 42042, 25740, 6435, 45, 990, 9240, 45045, 126126, 210210, 205920, 109395, 24310, 55, 1485, 17160

Offset: 1

Benoit Cloitre, Mar 20 2004

First column = A000217, second column = A050534, main diagonal = A001700, second diagonal = A033876.

Cf. A063007.

Table[(Binomial[n+k,k]Binomial[n,n-k])/2,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Dec 29 2021 *)

T(n,k)=(1/2)*binomial(n+k,k)*binomial(n,n-k)

Definition corrected by Harvey P. Dale, Dec 29 2021

cp's OEIS Frontend

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