A120259 -id:A120259 - cp's OEIS frontend

A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 20, 4, 1, 1, 70, 175, 50, 5, 1, 1, 252, 1764, 980, 105, 6, 1, 1, 924, 19404, 24696, 4116, 196, 7, 1, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1

Offset: 0

Paul Barry, Jun 13 2006

Columns are the central coefficients of the triangles T(n, k;r) with T(n, k;r)=Product{j=0..r, C(n+j, k+j)/C(n-k+j, j)}*[k<=n]; (r=0,A007318), (r=1;A001263),(r=2,A056939),(r=3,A056940),(r=4,A056941). Essentially A103905 as a number triangle with an extra diagonal of 1's. Central coefficients T(2n, n) are A008793. Row sums are A120259. Diagonal sums are A120260.

			Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   6,     3,     1;
  1,  20,    20,     4,    1;
  1,  70,   175,    50,    5,   1;
  1, 252,  1764,   980,  105,   6, 1;
  1, 924, 19404, 24696, 4116, 196, 7, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened

Row sums give A120259.

Cf. A000891, A000984, A103905, A120257.

T(n, k) = prod(j=0, k-1, binomial(2*n-2*k+j, n-k)/binomial(n-k+j, j)); \\ Seiichi Manyama, Apr 02 2021

Number triangle T(n, k)=[k<=n]*Product{j=0..k-1, C(2n-2k+j, n-k)/C(n-k+j, j)}

As a square array, this is T(n,m)=product{k=1..m, product{j=1..n, product{i=1..n, (i+j+k-1)/(i+j+k-2)}}}; - Paul Barry, May 13 2008

A343032 Row sums of triangle A073165.

Original entry on oeis.org

1, 2, 4, 9, 24, 78, 313, 1557, 9606, 73482, 696736, 8187149, 119214337, 2150935400, 48085463503, 1331903411529, 45708405952786, 1943464419169294, 102378212255343442, 6681679619583450775, 540264005909352759970, 54120992439329583459008, 6716802027097934788929023

Offset: 0

Seiichi Manyama, Apr 03 2021

nonn

Cf. A073165, A120259.

Table[Sum[Product[(n - k + i + j - 1)/(i + j - 1), {i, 1, k}, {j, 1, i}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2021 *)
Table[Sum[BarnesG[k+1] / BarnesG[n+1] * Sqrt[Gamma[k+1] * Gamma[(n-k+2)/2] * BarnesG[n-k+1] * BarnesG[n+k+2] / (Gamma[n-k+1] * Gamma[(n+k+2)/2] * BarnesG[2*k+2])], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2021 *)

a(n) = sum(k=0, n, prod(i=1, k, prod(j=1, i, (n-k+i+j-1)/(i+j-1))));

a(n) = Sum_{k=0..n} Product_{1<=i<=j<=k} (n-k+i+j-1)/(i+j-1).

Limit_{n->infinity} a(n)^(1/n^2) = 2^r * r^(r/2) * (1-r)^((1-r)/2) = 1.113022855718664043805172905388731078607920794227951582456470883692074109..., where r = 0.62986938372832785012478891433662812255632994055776040984266... is the root of the equation 2^(4*r) * (1-r)^(1-r) * r^(2*r) = (1+r)^(1+r). - Vaclav Kotesovec, Apr 03 2021

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A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

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