A283272 -id:A283272 - cp's OEIS frontend

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

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).

A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -512, -19171, -111645, 8255899, 287477144, 3248973702, -56353404842, -2946880278857, -50078654012311, -24091665240825, 19437354184565824, 486126425619195338, 4607922953609319032, -63107867988829247005, -3101395214088243725145

Offset: 0

Seiichi Manyama, Mar 05 2017

sign

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

Column k=9 of A283272.
Cf. A023878 (exp( Sum_{n>=1} sigma_10(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), A283339 (k=9), this sequence (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^9).

a(n) = -(1/n)*Sum_{k=1..n} sigma_10(k)*a(n-k).

A294587 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -1, 1, -1, -8, -5, -1, 1, -1, -16, -19, -3, 5, 1, -1, -32, -65, -13, 23, 1, 1, -1, -64, -211, -63, 131, 44, 13, 1, -1, -128, -665, -301, 815, 497, 104, 4, 1, -1, -256, -2059, -1383, 5195, 4840, 1149, 70, 0, 1, -1, -512, -6305, -6133, 33143, 45021, 13752, 662, -93, 2

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
   -1, -1,  -1,  -1,   -1, ...
   -2, -4,  -8, -16,  -32, ...
   -1, -5, -19, -65, -211, ...
   -1, -3, -13, -63, -301, ...

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

Columns k=0..2 give A022661, A266964, A294588.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A283272.

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.

A281581 a(n) = (152^(2n+2) + 152^(n+2) + 52^(n+3)3^(n+1) - 245^(n+1))/120.

Original entry on oeis.org

1, 4, 21, 127, 807, 5179, 33111, 210067, 1321887, 8255899, 51225351, 316067107, 1941032367, 11873549419, 72394874391, 440204293747, 2670669533247, 16172309991739, 97779619272231, 590423692897987, 3561340764760527, 21462312506478859

Offset: 0

Seiichi Manyama, Mar 05 2017

Seiichi Manyama, Table of n, a(n) for n = 0..1285
Index entries for linear recurrences with constant coefficients, signature (17,-104,268,-240).

Row n=5 of A283272.

Table[(15*2^(2*n+2) + 15*2^(n+2) + 5*2^(n+3)*3^(n+1) - 24*5^(n+1)) / 120, {n, 0, 21}] (* Indranil Ghosh, Mar 05 2017 *)

a(n) = (15*2^(2*n+2) + 15*2^(n+2) + 5*2^(n+3)*3^(n+1) - 24*5^(n+1)) / 120;
for (n=0, 21, print1(a(n),", ")); \\ Indranil Ghosh, Mar 05 2017

def A281581(n): return (15*2**(2*n+2) + 15*2**(n+2) + 5*2**(n+3)*3**(n+1) - 24*5**(n+1)) / 120 # Indranil Ghosh, Mar 05 2017

def A281581(n)
  (0..n).map{|i| (15 * 2 ** (2 * i + 2) + 15 * 2 ** (i + 2) + 5 * 2 ** (i + 3) * 3 ** (i + 1) - 24 * 5 ** (i + 1)) / 120}
end

G.f.: ( 1-13*x+57*x^2-82*x^3 ) / ( (6*x-1)*(4*x-1)*(2*x-1)*(5*x-1) ). - R. J. Mathar, Mar 19 2017

a(n) = 6^n +2^(n-1)-5^n+4^n/2. - R. J. Mathar, Mar 19 2017

cp's OEIS Frontend

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

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A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

Views

Author

Keywords

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Crossrefs

Formula

A294587 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^(j^k).

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Formula

A281581 a(n) = (152^(2n+2) + 152^(n+2) + 52^(n+3)3^(n+1) - 245^(n+1))/120.

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Mathematica

PARI

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cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Formula

A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

Views

Author

Keywords

Links

Crossrefs

Formula

A294587 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^(j^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A281581 a(n) = (15*2^(2*n+2) + 15*2^(n+2) + 5*2^(n+3)*3^(n+1) - 24*5^(n+1))/120.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Ruby

Formula

A281581 a(n) = (152^(2n+2) + 152^(n+2) + 52^(n+3)3^(n+1) - 245^(n+1))/120.