A331792
Expansion of ((1 - 4*x)/sqrt(1 - 8*x + 4*x^2) - 1)/(6*x^2).
Original entry on oeis.org
1, 8, 57, 400, 2810, 19824, 140497, 999968, 7143966, 51206320, 368094122, 2652720096, 19159794004, 138658606688, 1005231020865, 7299082678336, 53074479789878, 386419850997552, 2816685368479342, 20553133273532000, 150120362670452076
Offset: 0
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a[n_] := Sum[3^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, May 05 2021 *)
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N=20; x='x+O('x^N); Vec(((1-4*x)/sqrt(1-8*x+4*x^2)-1)/(6*x^2))
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{a(n) = sum(k=0, n, 3^k*binomial(n+1, k)*binomial(n+1, k+1))}
A331793
Expansion of ((1 - 5*x)/sqrt(1 - 10*x + 9*x^2) - 1)/(8*x^2).
Original entry on oeis.org
1, 10, 87, 740, 6285, 53550, 458115, 3934600, 33913881, 293244050, 2542684463, 22101612780, 192530903461, 1680415209270, 14692052109915, 128653303453200, 1128147127156785, 9905115333850650, 87066787614156807, 766127762539955700, 6747880819438628541
Offset: 0
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a[n_] := Sum[4^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, May 05 2021 *)
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my(N=30, x='x+O('x^N)); Vec(((1-5*x)/sqrt(1-10*x+9*x^2)-1)/(8*x^2))
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a(n) = sum(k=0, n, 4^k*binomial(n+1, k)*binomial(n+1, k+1));
A331794
a(n) = Sum_{k=0..n} n^k * binomial(n+1,k) * binomial(n+1,k+1).
Original entry on oeis.org
1, 4, 33, 400, 6285, 120456, 2714173, 70129984, 2040655401, 65956468600, 2342384363561, 90607200956064, 3789863084012629, 170370561866229648, 8188781210421259365, 418938023982360898816, 22724122083014879989905, 1302374806940392958470104
Offset: 0
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Flatten[{1, Table[Sum[n^k * Binomial[n+1,k] * Binomial[n+1,k+1], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Jan 26 2020 *)
Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 26 2020 *)
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a(n) = sum(k=0, n, n^k*binomial(n+1, k)*binomial(n+1, k+1));
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a(n) = polcoef(2/(1-2*(n+1)*x+((n-1)*x)^2+(1-(n+1)*x)*sqrt(1-2*(n+1)*x+((n-1)*x)^2)), n);
A331795
a(n) = Sum_{k=0..n} (-n)^k * binomial(n+1,k) * binomial(n+1,k+1).
Original entry on oeis.org
1, 0, -3, 40, -515, 7056, -102935, 1554288, -22775319, 265497760, 586651461, -230587852560, 13426823564869, -637734419560224, 28594259589697425, -1264238490602458784, 56015489395280490385, -2499557487903373341888, 112150411888789509887053
Offset: 0
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Flatten[{1, Table[Sum[(-1)^k * n^k * Binomial[n+1,k] * Binomial[n+1,k+1], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Jan 26 2020 *)
Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, -n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 26 2020 *)
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a(n) = sum(k=0, n, (-n)^k*binomial(n+1, k)*binomial(n+1, k+1));
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a(n) = polcoef(2/(1+2*(n-1)*x+((n+1)*x)^2+(1+(n-1)*x)*sqrt(1+2*(n-1)*x+((n+1)*x)^2)), n);
Showing 1-4 of 4 results.