OFFSET
0,3
COMMENTS
The Kn11 triangle sums of A094638 are given by the terms of this sequence. For the definitions of this and other triangle sums see A180662. [Johannes W. Meijer, Apr 20 2011]
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..868
Giulio Cerbai, Anders Claesson, and Bruce E. Sagan, Self-modified difference ascent sequences, arXiv:2408.06959 [math.CO], 2024. See p. 15.
FORMULA
O.g.f.: A(x) = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1+x*(k*x+x+1) - x*(k*x + 2*x + 1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: 1/(x*Q(0)-1)/x^4 + (1+x-x^3)/x^4, where Q(k)= 1 - x/(1 - (k+1)*x - x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
Conjecture: log(a(n)) ~ n*log(n)/2 - n*(1 + log(2))/2. - Vaclav Kotesovec, Sep 18 2024
EXAMPLE
A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2x) + x^3*(1+x)*(1+2x)*(1+3x) +...
MATHEMATICA
nmax = 30; CoefficientList[Series[Sum[x^(2*k)*Pochhammer[1 + 1/x, k], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2024 *)
Table[Sum[(-1)^k * StirlingS1[n+1-k, n+1-2*k], {k, 0, (n+1)/2}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 18 2024 *)
PROG
(PARI) a(n)=polcoeff(sum(k=0, n, x^k*prod(j=0, k, 1+j*x+x*O(x^n))), n)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 28 2006
STATUS
approved