A121680
a(n) = [x^n] (1 + x*(1+x)^(n+1) )^(n+1).
Original entry on oeis.org
1, 2, 12, 76, 655, 6816, 81690, 1109816, 16782399, 278438740, 5016899833, 97368894756, 2021749249403, 44658312247290, 1044437050070340, 25757381769393392, 667470006331599523, 18119105978249333988
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^5 = 655, since
(1 + x*(1+x)^5 )^5 = 1 + 5*x + 35*x^2 + 160*x^3 + 655*x^4 +...
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Table[Sum[Binomial[n+1,k] * Binomial[(n+1)*k,n-k], {k,0,n+1}], {n,0,20}] (* Vaclav Kotesovec, Oct 02 2020 *)
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a(n)=sum(k=0,n+1,binomial(n+1,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))
A121681
a(n) = A121680(n)/(n+1) = [x^n] (1 + x*(1+x)^(n+1) )^(n+1) / (n+1).
Original entry on oeis.org
1, 1, 4, 19, 131, 1136, 11670, 138727, 1864711, 27843874, 456081803, 8114074563, 155519173031, 3189879446235, 69629136671356, 1609836360587087, 39262941548917619, 1006616998791629666, 27044968746461571213
Offset: 0
At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^5 /5 = 655/5 = 131, since
(1 + x*(1+x)^5 )^5 = 1 + 5*x + 35*x^2 + 160*x^3 + 655*x^4 +...
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Table[Sum[Binomial[n+1,k] * Binomial[(n+1)*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+1)*k,n-k))/(n+1)
Showing 1-3 of 3 results.
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