A360712
Expansion of Sum_{k>0} (k * x * (1 + k*x^k))^k.
Original entry on oeis.org
1, 5, 27, 272, 3125, 46915, 823543, 16781312, 387421218, 10000078125, 285311670611, 8916102153177, 302875106592253, 11112006865911623, 437893890381640625, 18446744074783358976, 827240261886336764177, 39346408075327943829273
Offset: 1
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a[n_] := DivisorSum[n, #^(#+n/#-1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+k*x^k))^k))
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a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d, n/d-1));
A360730
Expansion of Sum_{k>=0} (k * x * (1 + k*x^2))^k.
Original entry on oeis.org
1, 1, 4, 28, 272, 3368, 50768, 902397, 18481408, 428556075, 11099001600, 317544062217, 9946366838784, 338537433281448, 12441407233436672, 491002325860132371, 20710640842719301632, 929821866165431838038, 44270378887441746923520
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^(n-k)*binomial(n-2*k, k));
A360731
Expansion of Sum_{k>=0} (k * x * (1 + k*x^3))^k.
Original entry on oeis.org
1, 1, 4, 27, 257, 3141, 46899, 827639, 16855357, 389100834, 10040378183, 286386193685, 8947506702834, 303875954083536, 11146559606379269, 439178938765108083, 18497974976610341624, 829420114454360154295, 39445018962975879216867
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x^3))^k))
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a(n) = sum(k=0, n\4, (n-3*k)^(n-2*k)*binomial(n-3*k, k));
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));
A360018
Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^2))^k.
Original entry on oeis.org
1, 1, 4, 28, 288, 3854, 63104, 1220729, 27248128, 689446671, 19501121536, 609753349945, 20883798220800, 777529328875208, 31266494467227648, 1350520199148276667, 62360172065142341632, 3065369553470816704832, 159818389764050045894656
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+(k*x)^2))^k))
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a(n) = sum(k=0, n\3, (n-2*k)^n*binomial(n-2*k, k));
A360032
Expansion of Sum_{k>=0} (k * x * (1 + (k * x)^3))^k.
Original entry on oeis.org
1, 1, 4, 27, 257, 3189, 48843, 889079, 18730597, 447945714, 11983618199, 354519428597, 11490618543066, 404910044246256, 15412461332440829, 630199633730994675, 27548323149955792880, 1282044807268698303751, 63284535745130267484867
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+(k*x)^3))^k))
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a(n) = sum(k=0, n\4, (n-3*k)^n*binomial(n-3*k, k));
Showing 1-6 of 6 results.