A266941 -id:A266941 - cp's OEIS frontend

A296601 L.g.f.: -log(Product_{k>=1} (1 - kx^k)^k) = Sum_{n>=1} a(n)x^n/n.

1, 9, 28, 81, 126, 330, 344, 833, 973, 1754, 1332, 5034, 2198, 5658, 8688, 13313, 4914, 28779, 6860, 54106, 45752, 33482, 12168, 254954, 93751, 78906, 255880, 505698, 24390, 1510700, 29792, 1671169, 1791312, 647114, 2819544, 12637371, 50654, 2282346, 14779520, 34058298, 68922, 68084220

Offset: 1

Ilya Gutkovskiy, May 20 2018

nonn

			L.g.f.: L(x) = x + 9*x^2/2 + 28*x^3/3 + 81*x^4/4 + 126*x^5/5 + 330*x^6/6 + 344*x^7/7 + 833*x^8/8 + 973*x^9/9 + ...
exp(L(x)) = 1 + x + 5*x^2 + 14*x^3 + 42*x^4 + 103*x^5 + 289*x^6 + 690*x^7 + 1771*x^8 + 4206*x^9 + ... + A266941(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Index entries for sequences related to sums of divisors

Column k=2 of A308502.

Cf. A001157, A078308, A266941, A266964.

nmax = 42; Rest[CoefficientList[Series[-Log[Product[(1 - k x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 42; Rest[CoefficientList[Series[Sum[k^3 x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[d^(n/d + 2), {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>=1} k^3*x^k/(1 - k*x^k).
a(n) = Sum_{d|n} d^(n/d+2).
a(p) = p^3 + 1 where p is a prime.
From Seiichi Manyama, Jun 24 2019: (Start)
Suppose given two sequences f(n) and g(n), n>0, we define a new sequence a(n), n>0, by a(n) = Sum_{d|n} d*f(d)*g(d)^(n/d).
L.g.f.: -log(Product_{n>0} (1 - g(n)*x^n)^f(n)) = Sum_{n>0} a(n)*x^n/n. (See A266964.)
If we set f(n) = n and g(n) = n, we get this sequence. (End)

A294582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j)^j.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 13, 1, 1, 17, 36, 42, 24, 1, 1, 33, 98, 148, 103, 48, 1, 1, 65, 276, 546, 489, 289, 86, 1, 1, 129, 794, 2068, 2467, 1959, 690, 160, 1, 1, 257, 2316, 7962, 12969, 14281, 6326, 1771, 282, 1, 1, 513, 6818, 30988, 70243

Offset: 0

Seiichi Manyama, Nov 02 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   13, 42, 148, 546, 2068, ...

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

Columns k=0..2 give A000219, A266941, A285674.
Rows n=0-1 give A000012.
Cf. A292193, A294580.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.

A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 14, 1, 1, 17, 36, 42, 25, 1, 1, 33, 98, 140, 103, 56, 1, 1, 65, 276, 498, 481, 289, 97, 1, 1, 129, 794, 1844, 2419, 1774, 690, 198, 1, 1, 257, 2316, 7002, 12745, 12173, 5925, 1771, 354, 1, 1, 513, 6818, 27020, 69283, 89706, 56974, 20076, 4206, 672, 1

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   14, 42, 140, 498, 1844, ...

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

Columns k=0..3 give A006906, A266941, A285241, A294590.
Rows n=0-1 give A000012.
Cf. A144048, A294587.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k+1+j/d)) * A(n-j,k) for n > 0.

A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 7, 19, 71, 173, 583, 1443, 4255, 10648, 28929, 71159, 184740, 445626, 1110122, 2638328, 6369490, 14870194, 35031627, 80465028, 185556696, 419916149, 950785580, 2121471778, 4727971847, 10412230698, 22876886529, 49776871862, 107974178843, 232302695301

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5920

Cf. A000203, A006906, A266941, A318415, A006171, A061256, A318484.

nmax = 40; CoefficientList[Series[Product[1/(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]*(-1)^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[Series[1/s, {x, 0, nmax}], x]

a(n) ~ c * n^3 * 3^(n/3), where
c = 280631952508395331283883354935233682635.581151020... if mod(n,3)=0
c = 280631952508395331283883354935233682635.059082354... if mod(n,3)=1
c = 280631952508395331283883354935233682635.088610121... if mod(n,3)=2
In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(k)))/(18*(57 - 90*3^(1/3) + 35*3^(2/3)))) - Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(k)))/ ((-1)^(2*n/3)*(6*(3 + 2*(-3)^(1/3))^3*(-3 + (-3)^(2/3)))) - ((-1)^(1 - (4*n)/3)*Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(k))))/(486*(1 + (-1/3)^(1/3))* (1 - 2*(-1/3)^(2/3))^3)

cp's OEIS Frontend

A296601 L.g.f.: -log(Product_{k>=1} (1 - kx^k)^k) = Sum_{n>=1} a(n)x^n/n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j)^j.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A296601 L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^k) = Sum_{n>=1} a(n)*x^n/n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294582 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j)^j.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A318483 Expansion of Product_{k>=1} 1/(1 - k*x^k)^sigma(k), where sigma = A000203.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A296601 L.g.f.: -log(Product_{k>=1} (1 - kx^k)^k) = Sum_{n>=1} a(n)x^n/n.