A343573
a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 5, 28, 265, 3126, 46750, 823544, 16778257, 387420652, 10000015646, 285311670612, 8916100731047, 302875106592254, 11112006831322846, 437893890380906656, 18446744073843774497, 827240261886336764178, 39346408075300025340205
Offset: 1
Cf.
A023887,
A157019,
A157020,
A324158,
A324159,
A338661,
A339481,
A339482,
A339712,
A343567,
A343574.
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a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)
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a(n) = sumdiv(n, d, d^d*binomial(d+n/d-2, d-1));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-x^k))^k))
A339481
a(n) = Sum_{d|n} d^(n-d) * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 2, 2, 10, 2, 131, 2, 1282, 4376, 16907, 2, 1138272, 2, 5793475, 154455992, 469893122, 2, 49501130330, 2, 1318441711177, 19001093813466, 3138439911059, 2, 15989399214596398, 6675720214843752, 3937376603803099, 6754271297694102092, 47097064577536888014, 2
Offset: 1
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a[n_] := DivisorSum[n, #^(n - #) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
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a(n) = sumdiv(n, d, d^(n-d)*binomial(d+n/d-2, d-1));
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my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(k*x)^k))^k))
A339712
a(n) = Sum_{d|n} d^(d+n/d-1) * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 5, 28, 273, 3126, 46948, 823544, 16781441, 387421948, 10000078446, 285311670612, 8916102176891, 302875106592254, 11112006865913416, 437893890382064056, 18446744074783625217, 827240261886336764178, 39346408075327954053967, 1978419655660313589123980
Offset: 1
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a[n_] := DivisorSum[n, #^(# + n/# - 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)
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a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d+n/d-2, d-1));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-k*x^k))^k))
A339482
a(n) = Sum_{d|n} d^(n-d+1) * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 3, 4, 21, 6, 346, 8, 4617, 13132, 80696, 12, 4903847, 14, 40410966, 756336736, 2416181265, 18, 306560794753, 20, 6941876836216, 132964265599502, 34522735212626, 24, 116720277621236637, 33378601074218776, 51185893450298400, 60788365423272068968
Offset: 1
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a[n_] := DivisorSum[n, #^(n - # + 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
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a(n) = sumdiv(n, d, d^(n-d+1)*binomial(d+n/d-2, d-1));
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my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*(x/(1-(k*x)^k))^k))
A338689
a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-2, d-1).
Original entry on oeis.org
1, 3, 28, 223, 3126, 44660, 823544, 16514047, 387538588, 9951176994, 285311670612, 8903202187413, 302875106592254, 11107259264162760, 437894348359764856, 18444492187995996159, 827240261886336764178, 39345059356329821149097
Offset: 1
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a[n_] := DivisorSum[n, (-1)^(# - 1) * (n/#)^n * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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a(n) = sumdiv(n, d, (-1)^(d-1)*(n/d)^n*binomial(d+n/d-2, d-1));
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N=20; x='x+O('x^N); Vec(sum(k=1, N, (k*x/(1+(k*x)^k))^k))
A360832
Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^2) )^k.
Original entry on oeis.org
1, 1, 4, 28, 288, 3855, 63232, 1227291, 27511296, 699389444, 19880700928, 624817997477, 21512488648704, 805233062024021, 32556682898653184, 1413981749074790444, 65652661019642560512, 3245240681196968168619, 170146759140135777861632
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\2, (n-2*k)^n*binomial(n-k-1, k));
A360833
Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.
Original entry on oeis.org
1, 1, 4, 27, 257, 3189, 48843, 889080, 18731109, 448004763, 11987812504, 354763577414, 11503684020051, 405589341060930, 15447798292502206, 632069580794524857, 27649951709582591394, 1287748889361331630661, 63616184683123273364961
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\3, (n-3*k)^n*binomial(n-2*k-1, k));
A360831
Expansion of Sum_{k>0} (k * x)^k / (1 - (k * x)^k)^(k+1).
Original entry on oeis.org
1, 6, 30, 308, 3130, 49962, 823550, 17107464, 387617328, 10058609120, 285311670622, 8931600297696, 302875106592266, 11117432610599574, 437894531752211760, 18449277498826162192, 827240261886336764194, 39347911865350001626164
Offset: 1
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a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-(k*x)^k)^(k+1)))
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a(n) = sumdiv(n, d, d^n*binomial(d+n/d-1, d));
Showing 1-8 of 8 results.