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)));
A355473
Expansion of Sum_{k>=0} x^k/(1 - k^3 * x)^(k+1).
Original entry on oeis.org
1, 1, 3, 28, 497, 12736, 517297, 28793248, 2095968065, 199522773568, 23839495688321, 3482169003693304, 616298415199306369, 130134007837039167040, 32272959284595295173377, 9313050358489324003967176, 3101245112865402456422252033
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^3*x)^(k+1)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1+sum(k=1, N, exp(k^3*x)*x^k/k!)))
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a(n) = sum(k=0, n, k^(3*(n-k))*binomial(n, k));
A360935
Expansion of e.g.f. Sum_{k>=0} exp((k^k - 1)*x) * x^k/k!.
Original entry on oeis.org
1, 1, 1, 10, 159, 8306, 1346855, 801620870, 2064941077199, 20691706495244482, 1137052204448926181679, 255128692791512749880418782, 348784909594653094321340422905383, 2262992285674206001784964011734257207938
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1+x+sum(k=2, N, exp((k^k-1)*x)*x^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k^k-1)*x)^(k+1)))
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a(n) = sum(k=0, n, (k^k-1)^(n-k)*binomial(n, k));
Showing 1-3 of 3 results.