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

0,2

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.

Links

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

Formula

a(n, m) = Sum_{k=m,..,n} ( A055248(n, k) ), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in column m).

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<m.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A245180 A054506 A101026 * A125172 A344381 A073732

Adjacent sequences: A055246 A055247 A055248 * A055250 A055251 A055252

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, May 26 2000

Status

approved