A360728
Expansion of Sum_{k>=0} (k * x * (1 + x^3))^k.
Original entry on oeis.org
1, 1, 4, 27, 257, 3133, 46737, 824567, 16792845, 387700506, 10005766337, 285445919589, 8919587932524, 302975123887680, 11115145723728035, 438000897534309171, 18450681900124075166, 827395845674975999727, 39352977072147424071861
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+x^3))^k))
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a(n) = sum(k=0, n\4, (n-3*k)^(n-3*k)*binomial(n-3*k, k));
A360726
Expansion of Sum_{k>0} (k * x * (1 + x^k))^k.
Original entry on oeis.org
1, 5, 27, 264, 3125, 46741, 823543, 16778240, 387420570, 10000015625, 285311670611, 8916100729755, 302875106592253, 11112006831322817, 437893890380890625, 18446744073843770368, 827240261886336764177, 39346408075300025059665
Offset: 1
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a[n_] := DivisorSum[n, #^# * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+x^k))^k))
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a(n) = sumdiv(n, d, d^d*binomial(d, n/d-1));
A360775
Expansion of Sum_{k>=0} (x * (k + x^2))^k.
Original entry on oeis.org
1, 1, 4, 28, 260, 3152, 46913, 826677, 16823968, 388245283, 10016796672, 285699444297, 8926107792609, 303160590533808, 11120927427841820, 438196895219227683, 18457860168281435172, 827678295600605015006, 39364859979651634985089
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x*(k+x^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^(n-3*k)*binomial(n-2*k, k));
A368893
a(n) = Sum_{k=0..floor(n/3)} n^(n-2*k) * binomial(n-2*k,k).
Original entry on oeis.org
1, 1, 4, 30, 288, 3500, 51876, 908607, 18374656, 421492491, 10815040000, 306944040931, 9547373318400, 322972830958648, 11805432990665664, 463673398064821875, 19474259980847153152, 870954834559130974358, 41323803842611198131264
Offset: 0
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Join[{1}, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2 - n/2, -n/2}, -27/(4*n^2)], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 09 2024 *)
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a(n) = sum(k=0, n\3, n^(n-2*k)*binomial(n-2*k, k));
Showing 1-4 of 4 results.