A121674
a(n) = [x^n] (1 + x*(1+x)^n )^n.
Original entry on oeis.org
1, 1, 5, 28, 233, 2376, 28102, 379016, 5707025, 94439440, 1699067321, 32951077193, 684009742319, 15110032165151, 353485501643471, 8721374385748256, 226128389777924385, 6142306518887606112, 174311816444805024379
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^4 )^4 = 233, since
(1 + x*(1+x)^4 )^4 = 1 + 4*x + 22*x^2 + 76*x^3 + 233*x^4 +...
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Table[Sum[Binomial[n,k] * Binomial[n*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*k,n-k))
A121676
a(n) = [x^n] (1 + x*(1+x)^(n-1) )^(n+1).
Original entry on oeis.org
1, 2, 6, 32, 250, 2412, 27524, 360600, 5296050, 85805420, 1515794467, 28926900312, 591903009295, 12907255696636, 298428274844730, 7284351640977920, 187013495992710210, 5033669346061547724, 141643700005223732471
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^3 )^5 = 250, since
(1 + x*(1+x)^3 )^5 = 1 + 5*x + 25*x^2 + 85*x^3 + 250*x^4 +...
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a(n)=sum(k=0,n+1,binomial(n+1,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));
A121677
a(n) = A121676(n)/(n+1) = [x^n] (1 + x*(1+x)^(n-1) )^(n+1) / (n+1).
Original entry on oeis.org
1, 1, 2, 8, 50, 402, 3932, 45075, 588450, 8580542, 137799497, 2410575026, 45531000715, 921946835474, 19895218322982, 455271977561120, 11000793881924130, 279648297003419318, 7454931579222301709
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^3 )^5/5 = 250/5 = 50, since
(1 + x*(1+x)^3 )^5 = 1 + 5*x + 25*x^2 + 85*x^3 + 250*x^4 +...
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Flatten[{1,Table[Sum[Binomial[n+1,k] * Binomial[(n-1)*k,n-k] / (n+1), {k,0,n+1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jun 12 2015 *)
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a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial((n-1)*k,n-k))/(n+1)
Showing 1-4 of 4 results.
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