A360618
Expansion of Sum_{k>=0} (k * x * (1 + k*x))^k.
Original entry on oeis.org
1, 1, 5, 43, 515, 7950, 150086, 3349945, 86296849, 2519907605, 82249222661, 2967449372028, 117266100841668, 5037282382077353, 233701540415817409, 11645959855678136519, 620389246928233860127, 35181554115178393462386, 2116059351692554708911298
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^n, {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+k*x))^k))
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a(n) = sum(k=0, n\2, (n-k)^n*binomial(n-k, k));
A360727
Expansion of Sum_{k>=0} (k * x * (1 + x^2))^k.
Original entry on oeis.org
1, 1, 4, 28, 264, 3206, 47684, 839249, 17058688, 393216567, 10134918592, 288815780665, 9016571143680, 306027510946208, 11219450971161024, 441846991480590475, 18602901833071633792, 833832341625621777368, 39642569136740054367808
Offset: 0
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nmax = 20; CoefficientList[1 + Series[Sum[(k*x*(1 + x^2))^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*x*(1+x^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^(n-2*k)*binomial(n-2*k, k));
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));
A360774
Expansion of Sum_{k>=0} (x * (k + x))^k.
Original entry on oeis.org
1, 1, 5, 31, 284, 3390, 49878, 871465, 17620450, 404554997, 10394845097, 295485704544, 9205957047661, 311922101632409, 11419004058232897, 449146827324857447, 18889836751306735360, 845892838094616177138, 40182354573647684880446
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x*(k+x))^k))
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a(n) = sum(k=0, n\2, (n-k)^(n-2*k)*binomial(n-k, k));
A360647
Expansion of Sum_{k>=0} (k^2 * x * (1 + x))^k.
Original entry on oeis.org
1, 1, 17, 761, 67739, 10029956, 2226004406, 691381685259, 286255287677425, 152360721379689043, 101358756787489940837, 82408168580060017122144, 80396790074312939684672316, 92691781529853274368541343021
Offset: 0
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Flatten[{1, Table[Sum[Binomial[n-k,k] * (n-k)^(2*(n-k)), {k,0,n/2}], {n,1,30}]}] (* Vaclav Kotesovec, Feb 16 2023 *)
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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) = sum(k=0, n\2, (n-k)^(2*(n-k))*binomial(n-k, k));
Showing 1-6 of 6 results.