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A201365 E.g.f.: exp(x) / (5 - 4*exp(x)). 4
1, 5, 45, 605, 10845, 243005, 6534045, 204972605, 7348546845, 296387331005, 13282361478045, 654762261324605, 35211177242722845, 2051349014835939005, 128701394409842982045, 8651475271312083756605, 620334325261670875138845, 47259638324026516284867005 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..17.

FORMULA

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+k*x).

O.g.f.: A(x) = 1/(1 - 5*x/(1-4*x/(1 - 10*x/(1-8*x/(1 - 15*x/(1-12*x/(1 - 20*x/(1-16*x/(1 - 25*x/(1-20*x/(1 - ...)))))))))), a continued fraction.

a(n) = Sum_{k=0..n} (-1)^(n-k) * 5^k * Stirling2(n,k) * k!.

a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*5^k*4^(n-k). - Philippe Deléham, Nov 30 2011

a(n) ~ n! / (4*(log(5/4))^(n+1)) . - Vaclav Kotesovec, Jun 13 2013

a(n) = log(5/4) * int {x = 0..inf} (ceiling(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015

EXAMPLE

E.g.f.: E(x) = 1 + 5*x + 45*x^2/2! + 605*x^3/3! + 10845*x^4/4! + 243005*x^5/5! +...

O.g.f.: A(x) = 1 + 5*x + 45*x^2 + 605*x^3 + 10845*x^4 + 243005*x^5 +...

where A(x) = 1 + 5*x/(1+x) + 2!*5^2*x^2/((1+x)*(1+2*x)) + 3!*5^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*5^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...

MATHEMATICA

Table[Sum[(-1)^(n-k)*5^k*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 13 2013 *)

With[{nn=20}, CoefficientList[Series[Exp[x]/(5-4Exp[x]), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Jul 09 2015 *)

PROG

(PARI) {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(5 - 4*exp(x+x*O(x^n))), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}

{a(n)=sum(k=0, n, (-1)^(n-k)*5^k*Stirling2(n, k)*k!)}

CROSSREFS

Cf. A201366, A201367, A201368, A050353.

Sequence in context: A007696 A090136 A090356 * A112940 A294332 A085356

Adjacent sequences:  A201362 A201363 A201364 * A201366 A201367 A201368

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Nov 30 2011

STATUS

approved

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Last modified March 21 10:00 EDT 2019. Contains 321368 sequences. (Running on oeis4.)