A220955 -id:A220955 - cp's OEIS frontend

A242449 a(n) = Sum_{k=0..n} C(n,k) * (2k+1)^(2n+1).

1, 28, 3612, 1064480, 560632400, 462479403072, 550095467201728, 891290348282967040, 1887146395301619304704, 5058811707344107766328320, 16746136671945501439084657664, 67088193422344140016282100785152, 319900900946743851959321101768511488

Offset: 0

Vaclav Kotesovec, May 14 2014

Generally, for p>=1, a(n) = Sum_{k=0..n} C(n,k) * (p*k+1)^(p*n+1) is asymptotic to n^(p*n+1) * p^(p*n+1) * r^(p*n+3/2+1/p) / (sqrt(p+r-p*r) * exp(p*n) * (1-r)^(n+1/p)), where r = p/(p+LambertW(p*exp(-p))).

G. C. Greubel, Table of n, a(n) for n = 0..190

Cf. A242373, A220955, A242446, A072034, A007820, A221214, A213193, A195242.

Table[Sum[Binomial[n,k]*(2*k+1)^(2*n+1),{k,0,n}],{n,0,20}]

for(n=0,30, print1(sum(k=0,n, binomial(n,k)*(2*k+1)^(2*n+1)), ", ")) \\ G. C. Greubel, Nov 16 2017

a(n) ~ n^(2*n+1) * 2^(2*n+1) * r^(2*n+2) / (sqrt(2-r) * exp(2*n) * (1-r)^(n+1/2)), where r = 2/(2+LambertW(2*exp(-2))) = 0.901829091937052...

A221214 O.g.f.: Sum_{n>=0} (3n+1)^(3n+1) * exp(-(3n+1)^3x) * x^n / n!.

Original entry on oeis.org

1, 255, 395388, 1525953330, 10977340509135, 126827739333023274, 2148335345336441463090, 50163717301669569182864400, 1544377393328765493716910877185, 60615459491155396034172113103266025, 2954227738557038665136475801709196246304

Offset: 0

Paul D. Hanna, Feb 27 2013

nonn

			O.g.f.: A(x) = 1 + 255*x + 395388*x^2 + 1525953330*x^3 + 10977340509135*x^4 +...
where A(x) = exp(-x) + 4^4*x*exp(-4^3*x) + 7^7*exp(-7^3*x)*x^2/2! + 10^10*exp(-10^3*x)*x^3/3! + 13^13*exp(-13^3*x)*x^4/4! + 16^16*exp(-16^3*x)*x^5/5! +... is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..159

Cf. A220955, A213193, A217900, A007820, A242449.

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(3*k+1)^(3*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[3*n+1,n+k]*3^(n+k)*StirlingS2[n+k,n],{k,0,2*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*exp(-(3*k+1)^3*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*x^k/(1+(3*k+1)^3*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(3*k+1)^(3*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (3*k+1)^(3*k+1) * x^k / (1 + (3*k+1)^3*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (3*k+1)^(3*n+1).
a(n) ~ n^(2*n+1/2) * 3^(6*n+7/3) / (sqrt(2*Pi*(1-r)) * exp(2*n) * r^(n+1/3) * (3-r)^(2*n+1)), where r = -LambertW(-3*exp(-3)) = 0.1785606278779211... (see A226750 = -r) . - Vaclav Kotesovec, May 13 2014

A222525 O.g.f.: Sum_{n>=0} (2n+1)^(2n) * exp(-(2n+1)^2x) * x^n / n!.

Original entry on oeis.org

1, 8, 232, 12160, 929376, 93590784, 11709432064, 1751777730560, 305065968649728, 60623947402670080, 13538933075023376384, 3356940619048979988480, 915040828127405123420160, 271974910674004076827115520, 87543520972441760055430348800, 30337462571518006406505729884160

Offset: 0

Paul D. Hanna, Feb 24 2013

nonn

			O.g.f.: A(x) = 1 + 8*x + 232*x^2 + 12160*x^3 + 929376*x^4 + 93590784*x^5 +...
where
A(x) = exp(-x) + 3^2*exp(-3^2*x)*x + 5^4*exp(-5^2*x)*x^2/2! + 7^6*exp(-7^2*x)*x^3/3! + 9^8*exp(-9^2*x)*x^4/4! + 11^10*exp(-11^2*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..290

Cf. A220955, A217900, A007820, A242373.

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(2*k+1)^(2*n),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[2^k*Binomial[2*n,k]*StirlingS2[k,n],{k,n,2*n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k) * x^k / (1 + (2*k+1)^2*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n).
a(n) = Sum_{k=0..n} 2^(n+k) * binomial(2*n,n+k) * stirling2(n+k,n). - Vaclav Kotesovec, May 13 2014
a(n) ~ 2^(4*n) * n^(n-1/2) / (sqrt(Pi*r*(1-r)) * exp(n) * (r*(2-r))^n), where r = -LambertW(-2*exp(-2)) = 0.4063757399599... (see A226775 = -r). - Vaclav Kotesovec, May 13 2014

A213193 O.g.f.: Sum_{n>=0} (4n+1)^(4n+1) * exp(-(4n+1)^4x) * x^n / n!.

Original entry on oeis.org

1, 3124, 191757120, 49208861869440, 33030777426968816640, 45829974166034718596428800, 114009204539207742166715857223680, 462192193445890293982679086838571270144, 2851153321165202191241172917762717987236478976

Offset: 0

Paul D. Hanna, Mar 01 2013

nonn

From Vaclav Kotesovec, May 13 2014: (Start)
Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).
a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).
(End)

			O.g.f.: A(x) = 1 + 3124*x + 191757120*x^2 + 49208861869440*x^3 +...
where
A(x) = exp(-x) + 5^5*x*exp(-5^4*x) + 9^9*exp(-9^4*x)*x^2/2! + 13^13*exp(-13^4*x)*x^3/3! + 17^17*exp(-17^4*x)*x^4/4! + 21^21*exp(-21^4*x)*x^5/5! +...
is a power series in x with integer coefficients.

Cf. A222525, A220955, A221214, A217900, A007820, A242449.

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(4*k+1)^(4*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[4*n+1,n+k]*4^(n+k)*StirlingS2[n+k,n],{k,0,3*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*exp(-(4*k+1)^4*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*x^k/(1+(4*k+1)^4*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(4*k+1)^(4*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (4*k+1)^(4*k+1) * x^k / (1 + (4*k+1)^4*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (4*k+1)^(4*n+1).
a(n) ~ n^(3*n+1/2) * 2^(16*n+9/2) / (sqrt(2*Pi*(1-r)) * exp(3*n) * r^(n+1/4) * (4-r)^(3*n+1)), where r = -LambertW(-4*exp(-4)) = 0.0793096051271136564391... . - Vaclav Kotesovec, May 13 2014

cp's OEIS Frontend

A242449 a(n) = Sum_{k=0..n} C(n,k) * (2*k+1)^(2*n+1).

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A221214 O.g.f.: Sum_{n>=0} (3*n+1)^(3*n+1) * exp(-(3*n+1)^3*x) * x^n / n!.

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A222525 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n) * exp(-(2*n+1)^2*x) * x^n / n!.

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A213193 O.g.f.: Sum_{n>=0} (4*n+1)^(4*n+1) * exp(-(4*n+1)^4*x) * x^n / n!.

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A242449 a(n) = Sum_{k=0..n} C(n,k) * (2k+1)^(2n+1).

A221214 O.g.f.: Sum_{n>=0} (3n+1)^(3n+1) * exp(-(3n+1)^3x) * x^n / n!.

A222525 O.g.f.: Sum_{n>=0} (2n+1)^(2n) * exp(-(2n+1)^2x) * x^n / n!.

A213193 O.g.f.: Sum_{n>=0} (4n+1)^(4n+1) * exp(-(4n+1)^4x) * x^n / n!.