A003659 Shifts left under Stirling2 transform. (Formerly M1681)

1, 1, 2, 6, 26, 152, 1144, 10742, 122772, 1673856, 26780972, 496090330, 10519217930, 252851833482, 6832018188414, 205985750827854, 6885220780488694, 253685194149119818, 10250343686634687424, 452108221967363310278, 21676762640915055856716

Offset

1,3

Comments

Apart from leading term, number of M-sequences from multicomplexes on at most 4 variables with no monomial of degree more than n+1.

Stirling2 transform of a(n) = [1, 1, 2, 6, 26, ...] is a(n+1) = [1, 2, 6, 26, ...].

Eigensequence of Stirling2 triangle A008277. - Philippe Deléham, Mar 23 2007

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

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

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 *)

Prog

(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

Crossrefs

Cf. A048801.

Cf. A153277, A153278. - Jonathan Vos Post, Dec 22 2008

Sequence in context: A185994 A032187 A372236 * A159602 A032271 A205502

Adjacent sequences: A003656 A003657 A003658 * A003660 A003661 A003662

Keyword

nonn,nice,eigen

Author

N. J. A. Sloane, Mira Bernstein

Status

approved