A294404 -id:A294404 - cp's OEIS frontend

A294402 E.g.f.: exp(-Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

1, -1, -3, -1, 1, 279, 301, 12263, 5601, -431281, -2140739, -77720721, -1755429983, -12569445721, 85768062381, -4458503862121, 43351731658561, 546719071653663, 31735514726673661, 291860504886837599, 5860390638855992001, 208620917963122666679

Offset: 0

Seiichi Manyama, Oct 30 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..448

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294403 (k=1), A294404 (k=2).

Cf. A000005, A018804, A028343, A294363.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j). - Ilya Gutkovskiy, Aug 17 2021

A294403 E.g.f.: exp(-Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, -1, -5, -7, 1, 839, 4171, 54305, 102817, -4303441, -74521349, -1595325271, -20768141855, -222701825737, 1485790534411, 65580347824529, 2880129557707201, 67631429234674655, 1543424936566399867, 23542870556917468889, 119940955037901088321

Offset: 0

Seiichi Manyama, Oct 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..439

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): A294402 (k=0), this sequence (k=1), A294404 (k=2).

Cf. A000203, A010815, A069097, A294361.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sigma(k)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j)^2. - Ilya Gutkovskiy, Aug 17 2021

A294951 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(-Sum_{j>=1} sigma_k(j) * x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, -3, 1, -1, -5, -1, 1, -1, -9, -7, 1, 1, -1, -17, -31, 1, 279, 1, -1, -33, -115, -23, 839, 301, 1, -1, -65, -391, -215, 3399, 4171, 12263, 1, -1, -129, -1267, -1319, 17519, 41311, 54305, 5601, 1, -1, -257, -3991, -6839, 102999, 387031, 473129, 102817, -431281

Offset: 0

Seiichi Manyama, Nov 12 2017

			Square array A(n,k) begins:
     1,   1,    1,     1,      1, ...
    -1,  -1,   -1,    -1,     -1, ...
    -3,  -5,   -9,   -17,    -33, ...
    -1,  -7,  -31,  -115,   -391, ...
     1,   1,  -23,  -215,  -1319, ...
   279, 839, 3399, 17519, 102999, ...

Columns k=0..2 give A294402, A294403, A294404.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000051(n+1).
Cf. A294947.

A(0,k) = 1 and A(n,k) = -(n-1)! * Sum_{j=1..n} j*sigma_k(j)*A(n-j,k)/(n-j)! for n > 0.

cp's OEIS Frontend

A294402 E.g.f.: exp(-Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A294403 E.g.f.: exp(-Sum_{n>=1} sigma(n) * x^n).

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Author

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PARI

Formula

A294951 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(-Sum_{j>=1} sigma_k(j) * x^j).

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