A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 16, 13, 5, 1, 1, 12, 30, 32, 19, 6, 1, 1, 16, 50, 71, 55, 26, 7, 1, 1, 20, 80, 140, 140, 86, 34, 8, 1, 1, 25, 120, 259, 316, 246, 126, 43, 9, 1, 1, 30, 175, 448, 660, 622, 399, 176, 53, 10, 1

Offset: 0

Author

Wolfdieter Lang, Apr 06 2001

Keywords

• nonn
• easy
• tabl

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is (1/(1-x*z/((1-z^2)*(1-z))))/(1-z).

Row sums give A052534. Column sequences (without leading zeros) give A000012 (powers of 1), A002620(n+1), A002624, A060099-A060101 for m=0..5.

The bisections of the column sequences give triangles A060102 and A060556.

Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - Paul Barry, Mar 28 2011

Examples

			p(3,x) = 1 + 4*x + 3*x^2 + x^3.
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  6,  8,  4,  1;
  1,  9, 16, 13,  5,  1;
  1, 12, 30, 32, 19,  6,  1;
  1, 16, 50, 71, 55, 26,  7,  1;
  ...
		

Links

• Vincenzo Librandi, Rows n = 0..100, flattened
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.

Crossrefs

Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101.

Cf. A060102, A060556, A188316.

Programs

Maple

A060098 := proc(n,k) add( binomial(n-2*i,n-2*i-k)*binomial(k+i-1,i), i=0..floor(n/2)) ; end proc:
seq(seq(A060098(n,k), k=0..n), n=0..12); # R. J. Mathar, Mar 29 2011
# Recurrence after Philippe Deléham:
T := proc(n, k) option remember;
if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else
T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, May 07 2023

Mathematica

t[n_, k_] := Sum[ Binomial[n-2*j, n-2*j-k]*Binomial[k+j-1, j], {j, 0, n/2}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

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

Number triangle T(n,k) = Sum_{i=0..floor(n/2)} C(n-2*i,n-2*i-k)*C(k+i-1,i). - Paul Barry, Mar 28 2011

From Philippe Deléham, Apr 20 2023: (Start)

T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise:

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k).

T(n, k) = A188316(n, k) + A188316(n-1, k). (End)