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

1, 5, 1, 19, 6, 1, 63, 25, 7, 1, 192, 88, 32, 8, 1, 552, 280, 120, 40, 9, 1, 1520, 832, 400, 160, 49, 10, 1, 4048, 2352, 1232, 560, 209, 59, 11, 1, 10496, 6400, 3584, 1792, 769, 268, 70, 12, 1, 26624, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 66304, 43520

Offset

0,2

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-z)^3)/(1-2*z)^4)/(1-x*z/(1-z)).

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.

The column sequences appear as A049612(n+1), A055585, A001794, A001789(n+3), A027608, A055586 for m=0..5.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

a(n, m) = Sum_{k=m..n} A055252(n, k).

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.

G.f. for column m: (((1-x)^3)/(1-2*x)^4)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([4, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

G.f.: (1 - x)^4/((1 - (1 + y)*x)*(1 - 2*x)^4). - Andrew Howroyd, Nov 13 2025

Example

Triangle begins:
  [0] 1
  [1] 5, 1
  [2] 19, 6, 1
  [3] 63, 25, 7, 1
  [4] 192, 88, 32, 8, 1
  [5] 552, 280, 120, 40, 9, 1
  [6] 1520, 832, 400, 160, 49, 10, 1
  [7] 4048, 2352, 1232, 560, 209, 59, 11, 1
Fourth row polynomial (n=3): p(3, x)= 63 + 25*x + 7*x^2 + x^3.

Maple

T := (n, k) -> binomial(n, k)*hypergeom([4, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

Prog

(PARI)
T(n) = [Vecrev(p) | p<-Vec((1 - x)^4/((1 - (1 + y)*x)*(1 - 2*x)^4) + O(x*x^n))];
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Nov 13 2025

Crossrefs

Row sums are A055589(n+1).

Cf. A007318, A055248, A055249, A055252.

Sequence in context: A297174 A226605 A389775 * A193861 A349544 A193857

Adjacent sequences: A055581 A055582 A055583 * A055585 A055586 A055587

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang, May 26 2000

Status

approved