A383897 Expansion of e.g.f. log(1 + x)/(1 - 2*x).
0, 1, 3, 20, 154, 1564, 18648, 261792, 4183632, 75345696, 1506551040, 33147751680, 795506123520, 20683638213120, 579135642946560, 17374156466688000, 555971699259648000, 18903058697617920000, 680509757426817024000, 25859377184592752640000, 1034374965738609696768000
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Magma
[0] cat [n le 1 select 1 else 2*n * Self(n-1) - (-1)^n * Factorial(n-1): n in [1..20]]; // Vincenzo Librandi, May 23 2025
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Mathematica
a[n_]:= n! * Sum[(-1)^(k-1)*2^(n-k)/k,{k,1,n}];Table[a[n],{n,0,19}] (* Vincenzo Librandi, May 23 2025 *) -
PARI
a(n) = n!*sum(k=1, n, (-1)^(k-1)*2^(n-k)/k);
Formula
a(n) = n! * Sum_{k=1..n} (-1)^(k-1) * 2^(n-k)/k.
a(n) = 2 * n * a(n-1) - (-1)^n * (n-1)!.
a(n) = (n+1) * a(n-1) + 2 * (n-1)^2 * a(n-2).
a(n) ~ log(3/2) * 2^n * n!. - Vaclav Kotesovec, May 23 2025
a(n) = n!*((-1)^(n+1)/n + (-1)^n*LerchPhi(-1/2,1,n) + 2^n*log(3/2)) for n > 0. - Stefano Spezia, May 23 2025