|
%I #14 Aug 25 2014 10:32:10
%S 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,
%T 112,104,63,26,7,1,1,128,256,272,192,96,34,8,1,1,256,576,688,552,321,
%U 138,43,9,1,1,512,1280,1696,1520,1002,501,190,53,10,1,1,1024,2816,4096
%N Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently.
%C 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))).
%C Column m (without leading zeros) is obtained from convolution of A000012 (powers of 1) with m-fold convoluted A011782.
%F a(n, m)= Am(n, 0) if n >= m >= 0 and a(n, m) := 0 if n<m; i.e. first column of the lower triangular matrix Am := prs^(m)(A007318) 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.
%F G.f. for column m: (1/(1-x))*(x*(1-x)/(1-2*x))^m, m >= 0.
%F T(n, k) = sum_{j=0..n-k} C(n-k, j)*C(k+j-1, k-1). - _Paul D. Hanna_, Jan 14 2004
%e {1}; {1, 1}; {1, 2, 1}; {1, 4, 3, 1}; {1, 8, 8, 4, 1}; ...
%e Fourth row polynomial (n=3): p(3,x)= 1+4*x+3*x^2+x^3
%t 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_ *)
%o (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 */
%o (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)
%Y 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.
%K easy,nonn,tabl
%O 0,5
%A _Wolfdieter Lang_, May 30 2000
|
