A143805 Eigensequence of triangle A130534.
1, 1, 2, 7, 36, 250, 2229, 24656, 329883, 5233837, 96907908, 2066551242, 50196458429, 1375782397859, 42203985613593, 1438854199059479, 54180508061067099, 2241000820010271224, 101316373253530824771, 4984697039955303538934, 265819807417517749652933
Offset: 0
Keywords
Examples
From _Paul D. Hanna_, May 20 2009: (Start) E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 36*x^4/4! + 250*x^5/5! + ... A(x) = 1 - log(1-x) + log(1-x)^2/2! - 2*log(1-x)^3/3! + 7*log(1-x)^4/4! - 36*log(1-x)^5/5! +- ... (End)
Links
- Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. [_N. J. A. Sloane_, Sep 16 2012]
Crossrefs
Cf. A143806.
Programs
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PARI
{a(n)=local(A=[1]);for(i=1,n,A=Vec(serlaplace(1+sum(k=1,#A,A[k]*(-log(1-x+x*O(x^n)))^k/k!))));A[n+1]} \\ Paul D. Hanna, May 20 2009 -
PARI
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)} {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(n-k-1)*Stirling1(n, k+1)*a(k)))} \\ Paul D. Hanna, Oct 01 2013
Formula
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) * Stirling1(n,k+1) * a(k) for n>0 with a(0)=1 (by definition). - Paul D. Hanna, Oct 01 2013
E.g.f.: Sum_{n>=0} a(n)*x^n/n! = 1 + Sum_{n>=1} a(n-1)*(-log(1-x))^n/n!. - Paul D. Hanna, May 20 2009
Conjecture: a(n) = R(n,0) where R(n,k) = R(n-1,n-1) + Sum_{j=0..k-1} (j+1)*R(n-1,j) for 0 <= k <= n with R(0,0) = 1. - Mikhail Kurkov, Jul 18 2025
Extensions
Extended by Paul D. Hanna, May 20 2009
Offset 0 by Georg Fischer, Apr 14 2024
Comments