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A055584 Triangle of partial row sums (prs) of triangle A055252.

%I A055584 #9 Sep 23 2024 23:02:23
%S A055584 1,5,1,19,6,1,63,25,7,1,192,88,32,8,1,552,280,120,40,9,1,1520,832,400,
%T A055584 160,49,10,1,4048,2352,1232,560,209,59,11,1,10496,6400,3584,1792,769,
%U A055584 268,70,12,1,26624,16896,9984,5376,2561,1037,338,82,13,1,66304,43520
%N A055584 Triangle of partial row sums (prs) of triangle A055252.
%C A055584 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)^3)/(1-2*z)^4)/(1-x*z/(1-z)).
%C A055584 This is the fourth 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 A055584 The column sequences appear as A049612(n+1), A055585, A001794, A001789(n+3), A027608, A055586 for m=0..5.
%F A055584 a(n, m)=sum(A055252(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in column m).
%F A055584 Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055252(n, m), n >= m >= 0, a(n, m) := 0 if n<m.
%F A055584 G.f. for column m: (((1-x)^3)/(1-2*x)^4)*(x/(1-x))^m, m >= 0.
%F A055584 T(n, k) = binomial(n, k)*hypergeom([4, k - n], [k + 1], -1). - _Peter Luschny_, Sep 23 2024
%e A055584 [0] 1
%e A055584 [1] 5, 1
%e A055584 [2] 19, 6, 1
%e A055584 [3] 63, 25, 7, 1
%e A055584 [4] 192, 88, 32, 8, 1
%e A055584 [5] 552, 280, 120, 40, 9, 1
%e A055584 [6] 1520, 832, 400, 160, 49, 10, 1
%e A055584 [7] 4048, 2352, 1232, 560, 209, 59, 11, 1
%e A055584 Fourth row polynomial (n=3): p(3, x)= 63 + 25*x + 7*x^2 + x^3.
%p A055584 T := (n, k) -> binomial(n, k)*hypergeom([4, k - n], [k + 1], -1):
%p A055584 for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # _Peter Luschny_, Sep 23 2024
%Y A055584 Cf. A007318, A055248, A055249, A055252. Row sums: A049600(n+1, 4).
%K A055584 easy,nonn,tabl
%O A055584 0,2
%A A055584 _Wolfdieter Lang_, May 26 2000