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

%I A055249 #14 Nov 03 2016 23:43:56
%S A055249 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,
%T A055249 30,8,1,576,448,321,201,102,38,9,1,1280,1024,769,522,303,140,47,10,1,
%U A055249 2816,2304,1793,1291,825,443,187,57,11,1,6144,5120,4097,3084,2116,1268,630
%N A055249 Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal's triangle A007318).
%C A055249 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)).
%C A055249 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.
%C A055249 The column sequences appear in A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 for m=0..7.
%H A055249 G. C. Greubel, <a href="/A055249/b055249.txt">Table of n, a(n) for n = 0..1274</a>
%F A055249 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).
%F A055249 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.
%F A055249 G.f. for column m: ((1-x)/(1-2*x)^2)*(x/(1-x))^m, m >= 0.
%F A055249 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
%e A055249 1;
%e A055249 3,1;
%e A055249 8,4,1;
%e A055249 20,12,5,1;
%e A055249 ...
%e A055249 Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3
%t A055249 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 *)
%Y A055249 Cf. A007318, A055248, A008949. Row sums: A049611(n+1) = A055252(n, 0).
%K A055249 nonn,tabl,easy
%O A055249 0,2
%A A055249 _Wolfdieter Lang_, May 26 2000