A121673
a(n) = [x^n] (1 + x*(1+x)^(n-1) )^n.
Original entry on oeis.org
1, 1, 3, 16, 131, 1306, 15257, 203967, 3047907, 50115310, 896746169, 17308420306, 357767229778, 7872926416538, 183537476164902, 4513828442107368, 116688468769638435, 3160881019508153238, 89471871451166037425
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^3 )^4 = 131, since
(1 + x*(1+x)^3 )^4 = 1 + 4*x + 18*x^2 + 52*x^3 + 131*x^4 +...
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Table[Sum[Binomial[n,k] * Binomial[(n-1)*k,n-k], {k,0,n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
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a(n)=sum(k=0,n,binomial(n,k)*binomial((n-1)*k,n-k))
A121675
a(n) = [x^n] (1 + x*(1+x)^(n+1) )^n.
Original entry on oeis.org
1, 1, 7, 43, 371, 3926, 47622, 654151, 9999523, 167557174, 3046387103, 59616689595, 1247357472869, 27747682830531, 653192297754076, 16206706672425167, 422358302959175123, 11526119161103900834
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^4 = 371, since
(1 + x*(1+x)^5 )^4 = 1 + 4*x + 26*x^2 + 104*x^3 + 371*x^4 +...
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Table[Sum[Binomial[n,k] * Binomial[(n+1)*k,n-k], {k,0,n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
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a(n)=sum(k=0,n,binomial(n,k)*binomial((n+1)*k,n-k))
A382859
a(n) = Sum_{k=0..n} binomial(n,k) * binomial((n-1)*(k+1),n-k).
Original entry on oeis.org
1, 1, 5, 37, 345, 3851, 49468, 713931, 11391985, 198523495, 3741919446, 75702725440, 1633591960883, 37404262517506, 904734768056239, 23030071358784701, 614912094171482849, 17172036245893988575, 500281954849350450946, 15170753984617328108901
Offset: 0
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[&+[Binomial(n,k) * Binomial((n-1)*(k+1),n-k): k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Apr 09 2025
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Table[Sum[Binomial[n,k] * Binomial[(n-1)*(k+1),n-k], {k,0,n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 07 2025 *)
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a(n) = sum(k=0, n, binomial(n, k)*binomial((n-1)*(k+1), n-k));
Showing 1-3 of 3 results.