A355463
Expansion of Sum_{k>=0} (x/(1 - k^k * x))^k.
Original entry on oeis.org
1, 1, 2, 10, 131, 5656, 869097, 490286392, 1264458639313, 12443651667592768, 681538604797281047489, 153070077563816488157872384, 205935348854901274982393017521537, 1352544986573612111579941739713633174912
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(k*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^k*x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k^(k*(n-k))*binomial(n-1, k-1)));
A355472
Expansion of Sum_{k>=0} (x/(1 - k^3 * x))^k.
Original entry on oeis.org
1, 1, 2, 18, 275, 6680, 258897, 13646776, 959706169, 88651586048, 10272048320897, 1462972094910224, 253355867842243905, 52387780870782231424, 12745274175326359046785, 3615579524073585972982544, 1184928928181459098548941633, 444427677344332049739011858432
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^3*x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k^(3*(n-k))*binomial(n-1, k-1)));
A360684
Expansion of Sum_{k>=0} (x * (1 + k^2 * x))^k.
Original entry on oeis.org
1, 1, 2, 9, 44, 308, 2391, 22851, 241570, 2937179, 39192998, 579482352, 9328260061, 162563246381, 3062996934322, 61499850730949, 1327236820161040, 30176760155713420, 733829463528115523, 18639130961053854975, 504241689606231891890
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-k,k] * (n-k)^(2*k), {k,0,n}], {n,1,30}]}] (* Vaclav Kotesovec, Feb 16 2023 *)
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k^2*x))^k))
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a(n) = sum(k=0, n\2, (n-k)^(2*k)*binomial(n-k, k));
Showing 1-3 of 3 results.