A318638
Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).
Original entry on oeis.org
1, 6, 27, 109, 405, 1467, 5103, 17550, 59050, 197100, 649539, 2126991, 6908733, 22325625, 71744625, 229602925, 731794257, 2324602206, 7360989291, 23245524600, 73222475256, 230128853031, 721764371007, 2259440202825, 7060738412026, 22029517662984, 68630377426119, 213516777941712, 663426981193869, 2058911488612863, 6382625094934119, 19765549255048254, 61149666233193318
Offset: 1
G.f.: A(x) = x + 6*x^2 + 27*x^3 + 109*x^4 + 405*x^5 + 1467*x^6 + 5103*x^7 + 17550*x^8 + 59050*x^9 + 197100*x^10 + 649539*x^11 + 2126991*x^12 + ...
such that
A(x) = x + (3 + x^2)^2 - 3^2 + (3 + x^3)^3 - 3^3 + (3 + x^4)^4 - 3^4 + (3 + x^5)^5 - 3^5 + (3 + x^6)^6 - 3^6 + (3 + x^7)^7 - 3^7 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 3:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...
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{a(n) = polcoeff( sum(m=1,n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}
for(n=1,100, print1(a(n),", "))
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a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021
A338685
a(n) = Sum_{d|n} d^n * binomial(d, n/d).
Original entry on oeis.org
1, 8, 81, 1040, 15625, 282123, 5764801, 134610944, 3486804084, 100097656250, 3138428376721, 107025924222976, 3937376385699289, 155582338242342053, 6568408660888671875, 295155786482995691520, 14063084452067724991009, 708240750793407501694308, 37589973457545958193355601
Offset: 1
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a[n_] := DivisorSum[n, #^n * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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a(n) = sumdiv(n, d, d^n*binomial(d, n/d));
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N=20; x='x+O('x^N); Vec(sum(k=1, N, (1+(k*x)^k)^k-1))
A338694
a(n) = Sum_{d|n} d^d * binomial(d, n/d).
Original entry on oeis.org
1, 8, 81, 1028, 15625, 280017, 5764801, 134219264, 3486784428, 100000031250, 3138428376721, 106993206079936, 3937376385699289, 155568095575106627, 6568408355712921875, 295147905179822588160, 14063084452067724991009, 708235345355351624428356, 37589973457545958193355601
Offset: 1
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a[n_] := DivisorSum[n, #^# * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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a(n) = sumdiv(n, d, d^d*binomial(d, n/d));
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N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+k*x^k)^k-k^k))
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));
A360759
a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).
Original entry on oeis.org
1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153
Offset: 1
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a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
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a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))
A360770
Expansion of Sum_{k>0} (x * (k + x^k))^k.
Original entry on oeis.org
1, 5, 27, 260, 3125, 46684, 823543, 16777472, 387420498, 10000003125, 285311670611, 8916100495009, 302875106592253, 11112006826381559, 437893890380860625, 18446744073726328848, 827240261886336764177, 39346408075296925015353
Offset: 1
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a[n_] := DivisorSum[n, #^(# - n/# + 1) * 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, (x*(k+x^k))^k))
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a(n) = sumdiv(n, d, d^(d-n/d+1)*binomial(d, n/d-1));
Showing 1-6 of 6 results.