A283340 -id:A283340 - cp's OEIS frontend

A283272 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is expansion of Product_{j>=1} (1-x^j)^(j^k) in power of x.

1, 1, -1, 1, -1, -1, 1, -1, -2, 0, 1, -1, -4, -1, 0, 1, -1, -8, -5, 0, 1, 1, -1, -16, -19, -1, 4, 0, 1, -1, -32, -65, -9, 21, 4, 1, 1, -1, -64, -211, -55, 127, 49, 7, 0, 1, -1, -128, -665, -285, 807, 500, 81, 3, 0, 1, -1, -256, -2059, -1351, 5179, 4809, 1038, 45

Offset: 0

Seiichi Manyama, Mar 04 2017

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
  -1, -2, -4,  -8, -16,  -32, ...
   0, -1, -5, -19, -65, -211, ...
   0,  0, -1,  -9, -55, -285, ...
   1,  4, 21, 127, 807, 5179, ...

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

Columns k=0-9 give A010815, A073592, A283263, A283264, A283271, A283336, A283337, A283338, A283339, A283340.
Row k=5 gives A281581.
Main diagonal gives A283333.
Cf. A144048.

G.f. of column k: Product_{j>=1} (1-x^j)^(j^k).

A283336 Expansion of exp( Sum_{n>=1} -sigma_6(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -32, -211, -285, 5179, 44784, 162062, -125122, -5187417, -32587255, -95706881, 122837972, 3039216222, 17745876032, 52825817007, -24340390929, -1256623249600, -7805634068163, -26364952524572, -20649978457115, 368666542515083, 2777231006764690

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=5 of A283272.
Cf. A023874 (exp( Sum_{n>=1} sigma_6(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), this sequence (k=6), A283337 (k=7), A283338 (k=8), A283339 (k=9), A283340 (k=10).

a[n_] := If[n<1, 1,-(1/n) * Sum[DivisorSigma[6, k] a[n - k], {k, n}]]; Table[a[n], {n, 0, 22}] (* Indranil Ghosh, Mar 16 2017 *)

a(n) = if(n<1, 1, -(1/n) * sum(k=1, n, sigma(k, 6) * a(n - k)));
for(n=0, 22, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 16 2017

G.f.: Product_{n>=1} (1 - x^n)^(n^5).

a(n) = -(1/n)*Sum_{k=1..n} sigma_6(k)*a(n-k).

A283337 Expansion of exp( Sum_{n>=1} -sigma_7(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -64, -665, -1351, 33111, 408149, 1959491, -4502590, -149420286, -1182474566, -3678670450, 22384197409, 377982157035, 2474860645111, 6161653683590, -48899064011245, -695916857379611, -4275491639488601, -10750056317745704, 69316545348329853

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=6 of A283272.
Cf. A023875 (exp( Sum_{n>=1} sigma_7(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), this sequence (k=7), A283338 (k=8), A283339 (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^6).

a(n) = -(1/n)*Sum_{k=1..n} sigma_7(k)*a(n-k).

A283338 Expansion of exp( Sum_{n>=1} -sigma_8(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -128, -2059, -6069, 210067, 3664420, 23366098, -116899962, -4133365357, -41809923367, -125160180169, 2447495850838, 42931762306584, 321967686614676, 281683012498569, -23874414003295851, -318729240693402530, -1992572289343189863

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=7 of A283272.
Cf. A023876 (exp( Sum_{n>=1} sigma_8(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), this sequence (k=8), A283339 (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^7).

a(n) = -(1/n)*Sum_{k=1..n} sigma_8(k)*a(n-k).

A283339 Expansion of exp( Sum_{n>=1} -sigma_9(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -256, -6305, -26335, 1321887, 32565169, 276211695, -2659962750, -111341327890, -1454216029918, -3323783801026, 227018039015019, 4636828146319845, 39615489757794355, -132865771935151820, -9075288352543844755, -132703303201618610765

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=8 of A283272.
Cf. A023877 (exp( Sum_{n>=1} sigma_9(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), A283338 (k=8), this sequence (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^8).

a(n) = -(1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k).

cp's OEIS Frontend

A283272 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is expansion of Product_{j>=1} (1-x^j)^(j^k) in power of x.

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A283336 Expansion of exp( Sum_{n>=1} -sigma_6(n)*x^n/n ) in powers of x.

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A283337 Expansion of exp( Sum_{n>=1} -sigma_7(n)*x^n/n ) in powers of x.

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A283338 Expansion of exp( Sum_{n>=1} -sigma_8(n)*x^n/n ) in powers of x.

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Author

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Formula

A283339 Expansion of exp( Sum_{n>=1} -sigma_9(n)*x^n/n ) in powers of x.

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Author

Keywords

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Formula