A005012 Shifts one place left under 6th-order binomial transform.
1, 1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, 58332035387017, 1457666574447247, 38485034941511935, 1069787864231083297, 31213730550761076769, 953352927192964243879, 30406448846308128741847
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Robert Israel, Table of n, a(n) for n = 0..420
- 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
Crossrefs
Cf. A004211.
Programs
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GAP
List([0..20],n->Sum([0..n],m->6^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018
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Maple
seq(6^n*BellB(n,1/6), n = 0 .. 50); # Robert Israel, Oct 20 2015
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Mathematica
Table[6^n BellB[n, 1/6], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
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
Extensions
a(0)=1 inserted by Alois P. Heinz, Oct 20 2015