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A005012 Shifts one place left under 6th-order binomial transform.
(Formerly M4434)
18
1, 1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, 58332035387017, 1457666574447247, 38485034941511935, 1069787864231083297, 31213730550761076769, 953352927192964243879, 30406448846308128741847 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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 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]
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Bell polynomial
FORMULA
a(n) = sum((6^(n-m))*stirling2(n,m), m=0..n). stirling2(n,m)=A008277(n,m).
E.g.f.: exp((exp(6*x)-1)/6) satisfies A'(x)/A(x) = exp(6*x).
G.f.: T(0)/(x*(1-x)) -1/x, where T(k) = 1 - 6*x^2*(k+1)/( 6*x^2*(k+1) - (1-x-6*x*k)*(1-7*x-6*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
a(n) = 6^n * B(n,1/6) where B(n,x) is the Bell polynomial of degree n. - Vladimir Reshetnikov, Oct 20 2015
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 6*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 6^n * n^n * exp(n/LambertW(6*n) - 1/6 - n) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^n). - Vaclav Kotesovec, Jul 15 2021
MAPLE
seq(6^n*BellB(n, 1/6), n = 0 .. 50); # Robert Israel, Oct 20 2015
MATHEMATICA
Table[6^n BellB[n, 1/6], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
PROG
(GAP) List([0..20], n->Sum([0..n], m->6^(n-m)*Stirling2(n, m))); # Muniru A Asiru, Mar 20 2018
CROSSREFS
Cf. A004211.
Sequence in context: A199564 A225032 A124403 * A123784 A091695 A002882
KEYWORD
nonn,easy,eigen
AUTHOR
EXTENSIONS
a(0)=1 inserted by Alois P. Heinz, Oct 20 2015
STATUS
approved

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Last modified March 19 06:56 EDT 2024. Contains 370953 sequences. (Running on oeis4.)