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A055252 Triangle of partial row sums (prs) of triangle A055249.

%I A055252 #9 Sep 23 2024 23:02:29
%S A055252 1,4,1,13,5,1,38,18,6,1,104,56,24,7,1,272,160,80,31,8,1,688,432,240,
%T A055252 111,39,9,1,1696,1120,672,351,150,48,10,1,4096,2816,1792,1023,501,198,
%U A055252 58,11,1,9728,6912,4608,2815,1524,699,256,69,12,1,22784,16640,11520
%N A055252 Triangle of partial row sums (prs) of triangle A055249.
%C A055252 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)^2)/(1-2*z)^3)/(1-x*z/(1-z)).
%C A055252 This is the third 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 A055252 The column sequences appear as A049611(n+1), A001793, A001788, A055580, A055581, A055582, A055583 for m=0..6.
%F A055252 a(n, m)=sum(A055249(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in column m).
%F A055252 Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055249(n, m), n >= m >= 0, a(n, m) := 0 if n<m.
%F A055252 G.f. for column m: (((1-x)^2)/(1-2*x)^3)*(x/(1-x))^m, m >= 0.
%F A055252 T(n, k) = binomial(n, k)*hypergeom([3, k - n], [k + 1], -1). - _Peter Luschny_, Sep 23 2024
%e A055252 [0] 1
%e A055252 [1] 4, 1
%e A055252 [2] 13, 5, 1
%e A055252 [3] 38, 18, 6, 1
%e A055252 [4] 104, 56, 24, 7, 1
%e A055252 [5] 272, 160, 80, 31, 8, 1
%e A055252 [6] 688, 432, 240, 111, 39, 9, 1
%e A055252 [7] 1696, 1120, 672, 351, 150, 48, 10, 1
%e A055252 Fourth row polynomial (n = 3): p(3, x) = 38 + 18*x + 6*x^2 + x^3.
%p A055252 T := (n, k) -> binomial(n, k)*hypergeom([3, k - n], [k + 1], -1):
%p A055252 for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # _Peter Luschny_, Sep 23 2024
%Y A055252 Cf. A007318, A055248, A055249. Row sums: A049612(n+1)= A055584(n, 0).
%K A055252 easy,nonn,tabl
%O A055252 0,2
%A A055252 _Wolfdieter Lang_, May 26 2000