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)));
A355464
Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).
Original entry on oeis.org
1, 2, 4, 17, 210, 9217, 1399298, 811229225, 2071392232962, 20710319937493889, 1137259214532706572162, 255141201504146525745627265, 348787971214016591166179037803522, 2262996819897931095524655885144485185409
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^k*x)^(k+1)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*x^k/k!)))
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a(n) = sum(k=0, n, k^(k*(n-k))*binomial(n, k));
A355471
Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.
Original entry on oeis.org
1, 1, 2, 10, 77, 808, 11257, 196072, 4136897, 103755904, 3034193921, 101901347944, 3885951145969, 166605168800704, 7961498177012993, 420976047757358776, 24475992585921169553, 1556007778666449968128, 107625967130820901112833
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(2*(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^2*x))^k))
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a(n) = if(n==0, 1, sum(k=1, n, k^(2*(n-k))*binomial(n-1, k-1)));
A360696
Expansion of Sum_{k>=0} (x * (1 + k^k * x))^k.
Original entry on oeis.org
1, 1, 2, 9, 98, 3212, 428525, 165045051, 342128248388, 2522279110319003, 90930729844450829580, 17690430223837969605522024, 13516362920784209950583739768297, 79459280482613898608830749440741093093
Offset: 0
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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) = sum(k=0, n\2, (n-k)^(k*(n-k))*binomial(n-k, k));
Showing 1-4 of 4 results.