A055252 Triangle of partial row sums (prs) of triangle A055249.

1, 4, 1, 13, 5, 1, 38, 18, 6, 1, 104, 56, 24, 7, 1, 272, 160, 80, 31, 8, 1, 688, 432, 240, 111, 39, 9, 1, 1696, 1120, 672, 351, 150, 48, 10, 1, 4096, 2816, 1792, 1023, 501, 198, 58, 11, 1, 9728, 6912, 4608, 2815, 1524, 699, 256, 69, 12, 1, 22784, 16640, 11520

Offset: 0

Author

Wolfdieter Lang, May 26 2000

Keywords

• easy
• nonn
• tabl

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is (((1-z)^2)/(1-2*z)^3)/(1-x*z/(1-z)).

This is the third member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.

The column sequences appear as A049611(n+1), A001793, A001788, A055580, A055581, A055582, A055583 for m=0..6.

Examples

			[0] 1
[1] 4, 1
[2] 13, 5, 1
[3] 38, 18, 6, 1
[4] 104, 56, 24, 7, 1
[5] 272, 160, 80, 31, 8, 1
[6] 688, 432, 240, 111, 39, 9, 1
[7] 1696, 1120, 672, 351, 150, 48, 10, 1
Fourth row polynomial (n = 3): p(3, x) = 38 + 18*x + 6*x^2 + x^3.
		

Crossrefs

Cf. A007318, A055248, A055249. Row sums: A049612(n+1)= A055584(n, 0).

Programs

Maple

T := (n, k) -> binomial(n, k)*hypergeom([3, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

Formula

a(n, m)=sum(A055249(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055249(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((1-x)^2)/(1-2*x)^3)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([3, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

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