A155586 -id:A155586 - cp's OEIS frontend

A155587 Expansion of (1 + x*c(x))/(1 - x), where c(x) is the g.f. of A000108.

1, 2, 3, 5, 10, 24, 66, 198, 627, 2057, 6919, 23715, 82501, 290513, 1033413, 3707853, 13402698, 48760368, 178405158, 656043858, 2423307048, 8987427468, 33453694488, 124936258128, 467995871778, 1757900019102, 6619846420554

Offset: 0

Paul Barry, Jan 24 2009

Row sums of A155586.
Hankel transform is A057079(n+2).
From Petros Hadjicostas, Aug 03 2020: (Start)
To prove R. J. Mathar's conjecture, note that the o.g.f. of the sequence implies (Sum_{n >= 0} a(n)*x^n)*(1 - x) = 1 + x*c(x); i.e., a(0) + Sum_{n >= 1} (a(n) - a(n-1))*x^n = 1 + Sum_{n >= 1} C(n-1)*x^n, where C(n) = A000108(n) (Catalan numbers).
Thus, C(n-1) = a(n) - a(n-1) (for n >= 1), and hence C(n) = a(n+1) - a(n). Since 2*(2*n - 1)*C(n-1) = (n + 1)*C(n), we get (n + 1)*a(n+1) + (-5*n + 1)*a(n) + 2*(2*n - 1)*a(n-1) = 0. The last equation implies R. J. Mathar's conjecture. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Gerard Cohen and Jean-Pierre Flori, On a generalized combinatorial conjecture involving addition mod 2^k - 1, IACR, Report 2011/400.
Jean-Pierre Flori, Fonctions booléennes, courbes algébriques et multiplication complexe, Thesis, ParisTech, Feb 03 2012.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A000108, A014138, A057079, A155586.

Partial sums of A120588.

a155587 n = a155587_list !! n
a155587_list = scanl (+) 1 a000108_list  -- Reinhard Zumkeller, Mar 01 2013

CatalanNumber := n -> binomial(2*n, n)/(n+1):
a := n -> ((3 - I*sqrt(3)))/2 - CatalanNumber(n)*hypergeom([1, n+1/2], [n+2], 4):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Aug 04 2020

a(n) = 1 + Sum_{k=0..n-1} A000108(k).
Conjecture: n*a(n) + (6-5*n)*a(n-1) + 2*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 15 2011
a(n) = A014138(n-1) + 2 for n > 0. - Reinhard Zumkeller, Mar 01 2013 [Corrected by Petros Hadjicostas, Aug 03 2020]
a(n+1) - a(n) = A000108(n). - Petros Hadjicostas, Aug 04 2020
a(n) = ((3 - i*sqrt(3)))/2 - CatalanNumber(n)*hypergeom([1, n + 1/2], [n + 2], 4). - Peter Luschny, Aug 04 2020

A343233 Triangle read by rows: Riordan triangle T = (1 - x*c(x), x), with the generating function c of A000108 (Catalan).

Original entry on oeis.org

1, -1, 1, -1, -1, 1, -2, -1, -1, 1, -5, -2, -1, -1, 1, -14, -5, -2, -1, -1, 1, -42, -14, -5, -2, -1, -1, 1, -132, -42, -14, -5, -2, -1, -1, 1, -429, -132, -42, -14, -5, -2, -1, -1, 1, -1430, -429, -132, -42, -14, -5, -2, -1, -1, 1

Offset: 0

Gary W. Adamson and Wolfdieter Lang, Apr 12 2021

As an unsigned sequence a(n) this is identical with the one of A155586(n+1), for n >= 0, but the triangle is not a simple signed version of A155586. See the formula.

This lower triangular Riordan matrix T of Toeplitz type is the inverse of the Riordan matrix (c(x), x) = |A106270|, also of Toeplitz type.

			The triangle matrix T begins:
  n/m     0    1    2   3   4   5   6   7   8   9 ...
  --------------------------------------------------
  0:      1
  1:     -1    1
  2:     -1   -1    1
  3:     -2   -1   -1   1
  4:     -5   -2   -1  -1   1
  5:    -14   -5   -2  -1  -1   1
  6:    -42  -14   -5  -2  -1  -1   1
  7:   -132  -42  -14  -5  -2  -1  -1   1
  8:   -429 -132  -42 -14  -5  -2  -1  -1   1
  9:  -1430 -429 -132 -42 -14  -5  -2  -1  -1   1
  ...

Cf. A106270 (unsigned), A155586.

The lower triangular matrix T satisfies: T = I - L^{tr}*|A106270|, also for the finite N X N version, with the unit matrix I and the lower triangular matrix L^{tr}(i, j) = delta_{i, j-1} (Kronecker symbol delta) with first lower diagonal of 1s and 0 otherwise.
T(n, n) = 1, and for T(n, m) = -C_{n - 1 - m } = - |A106270(n-1, m)|, for 0 <= m <= n-1, with the Catalan numbers C(n) = A000108, and T(n, m) = 0 for n < m.
O.g.f. of column m: (1/c(x))*x^m = (1 - x*c(x))*x^m (Riordan matrix of Toeplitz type), with the o.g.f. c of A000108.
O.g.f. row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m, that is the o.g.f. of the triangle. G(z, x) = c(z)/(1 - x*z).

cp's OEIS Frontend

A155587 Expansion of (1 + x*c(x))/(1 - x), where c(x) is the g.f. of A000108.

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