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
A136635
Triangle, read by rows, where T(n,k) = C(n,k) * C(2^k*3^(n-k), n) for n>=k>=0.
Original entry on oeis.org
1, 3, 2, 36, 30, 6, 2925, 2448, 660, 56, 1663740, 1265004, 353430, 42504, 1820, 6774333588, 4368213360, 1114691760, 139915440, 8561520, 201376, 204208594169580, 106458751541142, 23004238451040, 2630276490960
Offset: 0
Triangle begins:
1;
3, 2;
36, 30, 6;
2925, 2448, 660, 56;
1663740, 1265004, 353430, 42504, 1820;
6774333588, 4368213360, 1114691760, 139915440, 8561520, 201376;
204208594169580, 106458751541142, 23004238451040, 2630276490960, 167150463480, 5562289824, 74974368; ...
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Flatten[Table[Binomial[n,k]Binomial[2^k 3^(n-k),n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Dec 13 2012 *)
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{T(n,k)=binomial(n,k)*binomial(2^k*3^(n-k),n)}
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/* Using g.f.: */ {T(n,k)=polcoeff(polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x*y)^i/i!),n,x),k,y)}
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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