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A032033 Stirling transform of A032031. 14
1, 3, 21, 219, 3045, 52923, 1103781, 26857659, 746870565, 23365498683, 812198635941, 31055758599099, 1295419975298085, 58538439796931643, 2848763394161128101, 148537065755389540539, 8261178848690959117605 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also "AIJ" (ordered, indistinct, labeled) transform of 3,3,3,3...

Third row of array A094416 (generalized ordered Bell numbers).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution, arXiv:quant-ph/0303030, J. Phys. A.: Math. Gen  36 (2003) L273.

C. G. Bower, Transforms (2)

Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

FORMULA

E.g.f.: 1/(4-3*e^x).

a(n) = 3*A050352(n), n > 0.

a(n) = Sum_{k=0..n} Stirling2(n, k)*(3^k)*k!.

a(n) = (1/4)*Sum_{k>=0} k^n*(3/4)^k. - Karol A. Penson, Jan 25 2002

a(n) = Sum_{k=0..n} A131689(n,k)*3^k. - Philippe Deléham, Nov 03 2008

G.f. A(x)=B(x)/x, where B(x)=x+3*x^2+21*x^3+... = Sum_{n>=1} b(n)*x^n satisfies 4*B(x)-x = 3*B(x/(1-x)), and b(n)=3*Sum_{k=1..n-1} binomial(n-1,k-1)*b(k), b(1)=1. - Vladimir Kruchinin, Jan 27 2011

a(n) = log(4/3)*Integral_{x = 0..inf} (floor(x))^n * (4/3)^(-x) dx. - Peter Bala, Feb 14 2015

a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020

MATHEMATICA

a[n_] := PolyLog[-n, 3/4]/4; a[0] = 1; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 14 2011 *)

t = 30; Range[0, t]! CoefficientList[Series[1/(4 - 3 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

PROG

(PARI) a(n)=ceil(polylog(-n, 3/4)/4) \\ Charles R Greathouse IV, Jul 14 2014

CROSSREFS

Cf. A032031, A094416, A094418, A094419.

Sequence in context: A158838 A236963 A107716 * A218494 A099121 A107864

Adjacent sequences:  A032030 A032031 A032032 * A032034 A032035 A032036

KEYWORD

nonn,easy

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified February 26 20:15 EST 2021. Contains 341643 sequences. (Running on oeis4.)