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)
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))
A121678
a(n) = [x^n] (1 + x*(1+x)^n )^(n+1).
Original entry on oeis.org
1, 2, 9, 52, 425, 4236, 49294, 655096, 9731313, 159114880, 2832245911, 54400757016, 1119436524947, 24532373640334, 569732648555295, 13962373137304496, 359767723241891425, 9715902692094061488
Offset: 0
At n=5, a(5) = [x^5] (1 + x*(1+x)^5)^6 = 4236, since
(1+x*(1+x)^5)^6 = 1 + 6*x + 45*x^2 + 230*x^3 + 1050*x^4 + 4236*x^5 +...
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a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial(n*k,n-k))
A121679
a(n) = A121678(n)/(n+1) = [x^n] (1 + x*(1+x)^n )^(n+1) / (n+1).
Original entry on oeis.org
1, 1, 3, 13, 85, 706, 7042, 81887, 1081257, 15911488, 257476901, 4533396418, 86110501919, 1752312402881, 37982176570353, 872648321081531, 21162807249523025, 539772371783003416, 14433746294326451095
Offset: 0
At n=5, a(5) = [x^5] (1 + x*(1+x)^5)^6/6 = 4236/6 = 706, since
(1+x*(1+x)^5)^6 = 1 + 6*x + 45*x^2 + 230*x^3 + 1050*x^4 + 4236*x^5 +...
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Table[Sum[Binomial[n+1,k] * Binomial[n*k,n-k] / (n+1), {k,0,n+1}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
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a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial(n*k,n-k))/(n+1)
Showing 1-4 of 4 results.
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