A055587 Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 20, 13, 5, 1, 1, 32, 48, 38, 19, 6, 1, 1, 64, 112, 104, 63, 26, 7, 1, 1, 128, 256, 272, 192, 96, 34, 8, 1, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1, 1024, 2816, 4096

Offset: 0

Author

Wolfdieter Lang, May 30 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/((1-z)*(1-x*z*(1-z)/(1-2*z))).

Column m (without leading zeros) is obtained from convolution of A000012 (powers of 1) with m-fold convoluted A011782.

Examples

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

Crossrefs

Cf. A049600, column sequences are A000012 (powers of 1), A000079 (powers of 2), A001792, A049611, A049612, A055589, A055852-5 for m=0..9, row sums: A055588.

Programs

Mathematica

t[n_, k_] := Hypergeometric2F1[k, k-n, 1, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Paul D. Hanna *)

PARI

{T(n,k) = if( n<0 || k<0, 0, polcoeff( polcoeff( 1 / ((1 - z) * (1 - x*z * (1 - z) / (1 - 2*z) + z * O(z^n) + x * O(x^k))), k), n))}; /* Michael Somos, Sep 30 2003 */

PARI

{T(n,k)=if(k>n||n<0||k<0,0,if(k==0||k==n,1, sum(j=0,n-k,binomial(n-k,j)*binomial(k+j-1,k-1)););)} (Hanna)

Formula

a(n, m)= Am(n, 0) if n >= m >= 0 and a(n, m) := 0 if nA007318) with the lower triangular matrix A007318 (Pascal triangle) and prs^(m) is the partial row sums (prs) mapping for triangular matrices applied m times. See e.g. A055584 for m=4.

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

T(n, k) = sum_{j=0..n-k} C(n-k, j)*C(k+j-1, k-1). - Paul D. Hanna, Jan 14 2004