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A143805 Eigensequence of triangle A130534. 4
1, 1, 2, 7, 36, 250, 2229, 24656, 329883, 5233837, 96907908, 2066551242, 50196458429, 1375782397859, 42203985613593, 1438854199059479, 54180508061067099, 2241000820010271224, 101316373253530824771 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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=1..19.

M. 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

AUTHOR

Gary W. Adamson, Sep 01 2008

EXTENSIONS

Extended by Paul D. Hanna, May 20 2009

STATUS

approved

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Last modified February 23 16:00 EST 2020. Contains 332171 sequences. (Running on oeis4.)