A136636
a(n) = n * C(2*3^(n-1), n) for n>=1.
Original entry on oeis.org
2, 30, 2448, 1265004, 4368213360, 106458751541142, 19173684851378353296, 26413015283743616538733008, 285290979402099025600644272168880, 24601033850235942230699563821233785600080
Offset: 1
A136637
a(n) = Sum_{k=0..n} C(n, k) * C(2^k*3^(n-k), n).
Original entry on oeis.org
1, 5, 72, 6089, 3326498, 12405917044, 336474648380394, 69883583587428350874, 115099747754889610404191160, 1536533057081060754026861201898620, 168527150638482484315370462123098294514192
Offset: 0
More generally,
if Sum_{n>=0} log(1 + (p^n + r*q^n)*x )^n / n! = Sum_{n>=0} b(n)*x^n,
then b(n) = Sum_{k=0..n} C(n,k)*r^(n-k) * C(p^k*q^(n-k), n)
(a result due to _Vladeta Jovovic_, Jan 13 2008).
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Table[Sum[Binomial[n,k]*Binomial[2^k*3^(n-k),n], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
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{a(n)=sum(k=0,n,binomial(n,k)*binomial(2^k*3^(n-k),n))}
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/* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+(2^i+3^i)*x)^i/i!),n,x)}
A136638
a(n) = Sum_{k=0..[n/2]} C(n-k, k) * C(3^(n-2*k)*2^k, n-k).
Original entry on oeis.org
1, 3, 38, 2955, 1666194, 6775599252, 204212962736426, 47025953519744215608, 84798028785462127288681736, 1219731316443261012339196962784452, 141916030637329352970764084182705691263552
Offset: 0
More generally, if Sum_{n>=0} log(1 + b*p^n*x + d*q^n*x^2)^n/n! = Sum_{n>=0} a(n)*x^n then a(n) = Sum_{k=0..[n/2]} C(n-k,k)*b^(n-2k)*d^k*C(p^(n-2k)*q^k,n-k).
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Table[Sum[Binomial[n-k,k]*Binomial[2^k*3^(n-2*k),n-k], {k, 0, Floor[n/2]}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)
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{a(n)=sum(k=0,n\2,binomial(n-k,k)*binomial(3^(n-2*k)*2^k,n-k))}
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/* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x^2)^i/i!),n,x)}
Showing 1-3 of 3 results.
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