A055249 Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal's triangle A007318).

1, 3, 1, 8, 4, 1, 20, 12, 5, 1, 48, 32, 17, 6, 1, 112, 80, 49, 23, 7, 1, 256, 192, 129, 72, 30, 8, 1, 576, 448, 321, 201, 102, 38, 9, 1, 1280, 1024, 769, 522, 303, 140, 47, 10, 1, 2816, 2304, 1793, 1291, 825, 443, 187, 57, 11, 1, 6144, 5120, 4097, 3084, 2116, 1268, 630

Offset: 0

Author

Wolfdieter Lang, May 26 2000

Keywords

• nonn
• tabl
• easy

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)/(1-2*z)^2)/(1-x*z/(1-z)).

This is the second 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 in A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 for m=0..7.

Examples

			1;
3,1;
8,4,1;
20,12,5,1;
...
Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1274

Crossrefs

Cf. A007318, A055248, A008949. Row sums: A049611(n+1) = A055252(n, 0).

Programs

Mathematica

a[n_, m_] := Binomial[n, m]*Hypergeometric2F1[2, m-n, m+1, -1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2014 *)

Formula

a(n, m) = Sum_{k=m,..,n} ( A055248(n, k) ), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m) = Sum_{j=m,..,(n-1)} ( a(j, m) ) + A055248(n, m), n >= m >= 0, a(n, m) := 0 if n

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

a(n, m) = binomial(n, m) * 2F1(2, m-n; m+1; -1) where 2F1 is the hypergeometric function. Jean-François Alcover, Mar 11 2014

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