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URL: https://oeis.org/A143805

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A143805
Eigensequence of triangle A130534.
6
1, 1, 2, 7, 36, 250, 2229, 24656, 329883, 5233837, 96907908, 2066551242, 50196458429, 1375782397859, 42203985613593, 1438854199059479, 54180508061067099, 2241000820010271224, 101316373253530824771, 4984697039955303538934, 265819807417517749652933
OFFSET
0,3
COMMENTS
Triangle A130534 begins:
1;
1, 1;
2, 3, 1;
6, 11, 6, 1;
24, 50, 35, 10, 1;
...
Shift the entire triangle down 1 place, with T(0,0) = 1. Let T = the new triangle:
1;
1;
1, 1;
2, 3, 1;
...
This sequence is lim_{n -> infinity} T^n as a vector.
LINKS
Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. [N. J. A. Sloane, Sep 16 2012]
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
EXAMPLE
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)
PROG
(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
CROSSREFS
Cf. A143806.
Sequence in context: A007889 A125033 A034430 * A249637 A259793 A112293
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Sep 01 2008
EXTENSIONS
Extended by Paul D. Hanna, May 20 2009
Offset 0 by Georg Fischer, Apr 14 2024
STATUS
approved