A124380 O.g.f.: A(x) = Sum_{n>=0} x^n*Product_{k=0..n} (1 + k*x).
1, 1, 2, 4, 9, 22, 57, 157, 453, 1368, 4296, 13995, 47138, 163779, 585741, 2152349, 8113188, 31326760, 123748871, 499539900, 2058542819, 8651755865, 37054078481, 161591063250, 717032333816, 3235298221401, 14834735654080, 69085973044125
Offset: 0
Keywords
Examples
A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2x) + x^3*(1+x)*(1+2x)*(1+3x) +...
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.
Programs
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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 *) -
PARI
a(n)=polcoeff(sum(k=0,n,x^k*prod(j=0,k,1+j*x+x*O(x^n))),n)
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
Comments