A242449
a(n) = Sum_{k=0..n} C(n,k) * (2*k+1)^(2*n+1).
Original entry on oeis.org
1, 28, 3612, 1064480, 560632400, 462479403072, 550095467201728, 891290348282967040, 1887146395301619304704, 5058811707344107766328320, 16746136671945501439084657664, 67088193422344140016282100785152, 319900900946743851959321101768511488
Offset: 0
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Table[Sum[Binomial[n,k]*(2*k+1)^(2*n+1),{k,0,n}],{n,0,20}]
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for(n=0,30, print1(sum(k=0,n, binomial(n,k)*(2*k+1)^(2*n+1)), ", ")) \\ G. C. Greubel, Nov 16 2017
A222525
O.g.f.: Sum_{n>=0} (2*n+1)^(2*n) * exp(-(2*n+1)^2*x) * 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
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.
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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 *)
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{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), ", "))
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{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), ", "))
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{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), ", "))
Showing 1-2 of 2 results.
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