A384522 Expansion of e.g.f. 4/(7 - 3*exp(4*x)).
1, 3, 30, 426, 8040, 189768, 5375280, 177632976, 6708685440, 285038686848, 13456362881280, 698786099602176, 39586707755811840, 2429498408440009728, 160571526535426529280, 11370607719608891467776, 858870213271187908362240, 68928740686010010238353408
Offset: 0
Keywords
Links
- Wikipedia, Polylogarithm.
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[4/(7-3Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2025 *) -
PARI
a(n) = (-4)^(n+1)*polylog(-n, 7/3)/7;
Formula
a(n) = (-4)^(n+1)/7 * Li_{-n}(7/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 4^(n+1)/7 * Sum_{k>=0} k^n * (3/7)^k.
a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * k! * Stirling2(n,k).
a(n) = (3/7) * Sum_{k=0..n} 7^k * (-4)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 4^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 7 * Sum_{k=1..n-1} (-4)^(k-1) * binomial(n-1,k) * a(n-k).