A192065 -id:A192065 - cp's OEIS frontend

A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A061255, A061256, A006171.
Cf. A299069, A192065, A107742.
Cf. A294361, A294363.

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 9, 5, 17, 17, 38, 33, 88, 75, 169, 181, 343, 353, 712, 728, 1348, 1518, 2591, 2898, 5025, 5615, 9259, 10866, 17160, 20111, 31775, 37264, 57130, 68782, 102663, 123698, 183793, 221708, 323077, 395325, 566079, 693248, 987086, 1210110, 1700074, 2100674, 2915549

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Convolution of A192065 and A255528.

Weigh transform of A001065.

N. J. A. Sloane, Transforms

Cf. A000203, A001065, A192065, A255528, A318784.

with(numtheory): a:=series(mul((1+x^k)^(sigma(k)-k),k=1..100),x=0,47): seq(coeff(a,x,n),n=0..46); # Paolo P. Lava, Apr 02 2019

nmax = 46; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

G.f.: Product_{k>=1} (1 + x^k)^A001065(k).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*(sigma_1(d) - d) ) * x^k/k).
a(n) ~ exp(-Pi^4 / (864*(Pi^2 - 6)*Zeta(3)) - Pi^2 * n^(1/3) / (12*(2*(Pi^2 - 6)*Zeta(3))^(1/3)) + 3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (2^(17/24) * sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018

A316962 Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 1, 3, 7, 11, 25, 51, 87, 129, 286, 462, 760, 1312, 2102, 3470, 5988, 8840, 13884, 22577, 33545, 55961, 85341, 126705, 194317, 293621, 435040, 641472, 971503, 1462483, 2108161, 3124489, 4474579, 6545809, 9561923, 13518678, 19809034, 28387625, 40286631, 57039233

Offset: 0

Ilya Gutkovskiy, Jul 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000203, A180305, A192065, A279786, A316961.

with(numtheory): a:=series(mul(1+sigma(k)*x^k,k=1..100),x=0,39): seq(coeff(a,x,n),n=0..38); # Paolo P. Lava, Apr 02 2019

nmax = 38; CoefficientList[Series[Product[(1 + DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[1, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*sigma(j)^k*x^(j*k)/k).

A318484 Expansion of Product_{k>=1} (1 + k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 6, 18, 52, 142, 404, 1018, 2624, 6645, 16124, 38857, 92245, 214841, 494098, 1125062, 2522188, 5604930, 12327860, 26838595, 57913194, 123951482, 263019720, 553989989, 1158449522, 2405179547, 4961047246, 10168544537, 20714279168, 41952595411, 84494479578

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000203, A022629, A266891, A318416, A107742, A192065, A318483.

nmax = 40; CoefficientList[Series[Product[(1+k*x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k], j]*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[s, x]

A321877 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 7, 6, 1, 1, 9, 15, 14, 10, 1, 1, 17, 37, 41, 28, 17, 1, 1, 33, 99, 137, 107, 58, 25, 1, 1, 65, 277, 491, 487, 286, 106, 38, 1, 1, 129, 795, 1829, 2429, 1749, 700, 201, 59, 1, 1, 257, 2317, 6971, 12763, 12056, 5901, 1735, 372, 86

Offset: 0

Ilya Gutkovskiy, Nov 20 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   7,   15,   37,    99,    277,  ...
   6,  14,   41,  137,   491,   1829,  ...
  10,  28,  107,  487,  2429,  12763,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..9 give A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553.
Main diagonal gives A321042.
Cf. A321876.

Table[Function[k, SeriesCoefficient[Product[(1 + x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^(2 j))), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

G.f. of column k: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^k).

G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^(2*j)))).

A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 5, 0, 7, 3, 17, 0, 37, 6, 58, 23, 120, 21, 235, 67, 390, 161, 726, 230, 1349, 521, 2225, 1055, 3990, 1714, 7040, 3341, 11604, 6294, 20053, 10500, 34252, 19115, 56055, 34168, 94306, 56998, 157078, 99515, 254766, 171484, 419287, 283565

Offset: 0

Ilya Gutkovskiy, Oct 20 2019

nonn

Weigh transform of A048050.

Convolution of A326831 and A025147 is A319107. - Vaclav Kotesovec, Oct 26 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A048050, A192065, A319107, A326830.

with(numtheory):
g:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

nmax = 47; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, (-1)^(k/d + 1) d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

G.f.: Product_{k>=1} (1 + x^k)^A048050(k).

a(n) ~ exp(3*(2*(Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3) / (2^(7/3) * ((Pi^2 - 6)*Zeta(3))^(1/3)) - Pi^4 / (96*(Pi^2 - 6)*Zeta(3))) * 2^(19/24) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 26 2019

A328776 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} sigma(n) * x^n, where sigma = A000203.

Original entry on oeis.org

1, 3, 1, 3, -3, 2, -1, 4, 3, -8, -1, 6, 3, -4, -7, 12, 1, -6, 7, 0, -13, -13, 27, 13, -19, -11, 11, -21, -25, 191, -81, -300, 89, 327, 325, -745, -275, 579, -255, 1287, -453, -2075, -583, 2142, 5985, -6698, -6661, 6981, 3045, 3857, -7205, -2784, -5447, -4891, 48547

Offset: 1

Ilya Gutkovskiy, Oct 27 2019

sign

Inverse weigh transform of A000203.

Cf. A000203, A192065, A320780.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = DivisorSigma[1, n] - b[n, n - 1]; Array[a, 55]

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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

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A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

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A316962 Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

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A318484 Expansion of Product_{k>=1} (1 + k*x^k)^sigma(k), where sigma = A000203.

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A321877 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).

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A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

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A328776 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} sigma(n) * x^n, where sigma = A000203.

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