A360611
Expansion of Sum_{k>=0} (k * x * (1 + x))^k.
Original entry on oeis.org
1, 1, 5, 35, 341, 4230, 63844, 1135753, 23273363, 539881365, 13986073419, 400227436252, 12538263892232, 426810214125441, 15687071552060221, 619144491880324087, 26117514728711229877, 1172635546310430028562, 55833864788507320490268
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^(n-k), {k, 0, n/2}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 14 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0,N, (k*x*(1+x))^k))
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a(n) = sum(k=0,n\2, (n-k)^(n-k)*binomial(n-k, k));
A355496
Expansion of Sum_{k>=0} (k^3 * x/(1 - x))^k.
Original entry on oeis.org
1, 1, 65, 19812, 16836458, 30584805344, 101712712528352, 559155681922806328, 4726278437746021089208, 58187531579876705928027712, 1000523517685151396828602120640, 23235157037192774575979788565151104, 709057406693306876515431403267191583808
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x/(1-x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k^(3*k)*binomial(n-1, k-1)));
A355495
Expansion of Sum_{k>=0} (k^2 * x/(1 - x))^k.
Original entry on oeis.org
1, 1, 17, 762, 67772, 10032208, 2226273192, 691431572992, 286268594755712, 152365547943819264, 101361042063083269520, 82409537565402784477984, 80397802305461995791664944, 92692687015689239272783171264
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x/(1-x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k^(2*k)*binomial(n-1, k-1)));
A358741
Expansion of Sum_{k>=0} k! * ( k * x/(1 - x) )^k.
Original entry on oeis.org
1, 1, 9, 179, 6655, 400581, 35530421, 4357960999, 706230728379, 146116931998025, 37577989723572001, 11758017370126904091, 4398121660346674034039, 1938019214715102033590029, 993580299268226843514372045, 586357970017371399763899232271
Offset: 0
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nmax = 20; CoefficientList[1 + Series[Sum[k! * (k * x/(1 - x))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k*x/(1-x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k!*k^k*binomial(n-1, k-1)));
Showing 1-4 of 4 results.