A060086 -id:A060086 - cp's OEIS frontend

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

Wolfdieter Lang, Apr 06 2001

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

			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;
  ...

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.

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

Cf. A060102, A060556, A188316.

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

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 *)

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)

A181301 Number of 2-compositions of n having no column with equal entries. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

Original entry on oeis.org

1, 2, 6, 20, 64, 206, 662, 2128, 6840, 21986, 70670, 227156, 730152, 2346942, 7543822, 24248256, 77941648, 250529378, 805281526, 2588432308, 8320049072, 26743297998, 85961510758, 276307781200, 888141556360, 2854770939522

Offset: 0

Emeric Deutsch, Oct 12 2010

a(n)=A181299(n,0).

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.

Cf. A181299.

g := (1+z)*(1-z)^2/(1-3*z-z^2+z^3): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 27);

G.f. = (1+z)(1-z)^2/(1-3z-z^2+z^3).
a(n) = Sum_{k, 0<=k<=n} A060086(n,k)*2^k. - Philippe Deléham, Feb 24 2012
a(n) = 2*A033505(n-1), n>0. - R. J. Mathar, Jul 24 2022

cp's OEIS Frontend

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

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