A201354 Expansion of e.g.f. exp(x) / (4 - 3*exp(x)).

1, 4, 28, 292, 4060, 70564, 1471708, 35810212, 995827420, 31153998244, 1082931514588, 41407678132132, 1727226633730780, 78051253062575524, 3798351192214837468, 198049421007186054052, 11014905131587945490140, 650903915009792820650404, 40726453234725158535472348

Offset: 0

Author

Paul D. Hanna, Nov 30 2011

Keywords

• nonn
• easy

Examples

			E.g.f.: E(x) = 1 + 4*x + 28*x^2/2! + 292*x^3/3! + 4060*x^4/4! + 70564*x^5/5! + ...
O.g.f.: A(x) = 1 + 4*x + 28*x^2 + 292*x^3 + 4060*x^4 + 70564*x^5 + ...
where A(x) = 1 + 4*x/(1+x) + 2!*4^2*x^2/((1+x)*(1+2*x)) + 3!*4^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*4^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..370

Crossrefs

Cf. A000629, A032033, A050352, A123227, A136728.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(4*Exp(-x) -3) ))); // G. C. Greubel, Jun 08 2020

Maple

seq(coeff(series(1/(4*exp(-x) -3), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Mathematica

Table[Sum[(-1)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)

PARI

{a(n)=n!*polcoeff(exp(x+x*O(x^n))/(4 - 3*exp(x+x*O(x^n))), n)}

PARI

{a(n)=polcoeff(sum(m=0, n, 4^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)*4^k*Stirling2(n, k)*k!)}

Sage

[sum( (-1)^(n-j)*4^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

Formula

O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+k*x).

O.g.f.: A(x) = 1/(1 - 4*x/(1-3*x/(1 - 8*x/(1-6*x/(1 - 12*x/(1-9*x/(1 - 16*x/(1-12*x/(1 - 20*x/(1-15*x/(1 - ...))))))))))), a continued fraction.

a(n) = Sum_{k=0..n} (-1)^(n-k) * 4^k * Stirling2(n,k) * k!.

a(n) = 4*A050352(n) for n>0.

a(n) = Sum_{k=0..n} A123125(n,k)*4^k*3^(n-k). - Philippe Deléham, Nov 30 2011

a(n) = log(4/3) * Integral_{x = 0..oo} (ceiling(x))^n * (4/3)^(-x) dx. - Peter Bala, Feb 06 2015

G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 6*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

a(n) ~ n! / (3*(log(4/3))^(n+1)). - Vaclav Kotesovec, Jun 13 2013

a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020

From Seiichi Manyama, Nov 15 2023: (Start)

a(0) = 1; a(n) = -4*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = 4*a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

a(n) = (4/3)*A032033(n) - (1/3)*0^n. - Seiichi Manyama, Dec 21 2023