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

0,3

Comments

Triangle 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;...).

Sequence A143805 = lim_{n -> infinity} T^n as a vector.

Links

Table of n, a(n) for n=0..20.

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

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

Adjacent sequences: A143802 A143803 A143804 * A143806 A143807 A143808

Keyword

nonn,changed

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