A003659 Shifts left under Stirling2 transform.
1, 1, 2, 6, 26, 152, 1144, 10742, 122772, 1673856, 26780972, 496090330, 10519217930, 252851833482, 6832018188414, 205985750827854, 6885220780488694, 253685194149119818, 10250343686634687424, 452108221967363310278, 21676762640915055856716
Offset: 1
References
- S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..330
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. [_N. J. A. Sloane_, Sep 16 2012]
- Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047 [math.CO], 2008. [_Jonathan Vos Post_, Dec 22 2008]
- N. J. A. Sloane, Transforms
Crossrefs
Cf. A048801.
Programs
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Maple
stirtr:= proc(p) proc(n) add(p(k)*Stirling2(n,k), k=0..n) end end: a:= proc(n) option remember; `if`(n<3, 1, aa(n-1)) end: aa:= stirtr(a): seq(a(n), n=1..25); # Alois P. Heinz, Jun 22 2012 -
Mathematica
terms = 21; A[] = 0; Do[A[x] = Normal[Integrate[1 + A[Exp[x] - 1 + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x]*Range[0, terms]! // Rest (* Jean-François Alcover, May 23 2012, updated Jan 12 2018 *)
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PARI
{a(n)=local(A, E); if(n<0, 0, A=O(x); E=exp(x+x*O(x^n))-1; for(m=1, n, A=intformal( subst( 1+A, x, E+x*O(x^m)))); n!*polcoeff(A, n))} /* Michael Somos, Mar 08 2004 */ -
PARI
a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, stirling(i, j, 2)*v[j])); v; \\ Seiichi Manyama, Jun 24 2022
Formula
E.g.f. A(x) satisfies A(x)' = 1+A(exp(x)-1).
G.f. satisfies: Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - j*x)). - Ilya Gutkovskiy, May 09 2019
a(1) = 1; a(n+1) = Sum_{k=1..n} Stirling2(n,k) * a(k). - Seiichi Manyama, Jun 24 2022
Comments