A151954 -id:A151954 - cp's OEIS frontend

A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

1, 5, 10, 33, 26, 182, 50, 577, 811, 1750, 122, 16194, 170, 18982, 74900, 135425, 290, 847127, 362, 2498178, 4901060, 4209430, 530, 78564226, 9766251, 67138102, 387952660, 542674914, 842, 4866184552, 962, 8606778369, 31382832260, 17179953862, 6385992100, 422091411267, 1370, 274878038710

Offset: 1

Paul D. Hanna, Aug 04 2002

a(n) is the number of linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has two element marked (possibly equal). For instance, for n = 3, we have the following 10 linear partitions (where the marked elements are denoted by a and b, or by X when they coincide):

(X)(X)(X), (ab3), (a2b), (1ab), (ba3), (b2a), (1ba), (X23), (1X3), (12X). - Emanuele Munarini, Feb 03 2014

			a(10) = (10/1)^(2*1) +(10/2)^(2*2) +(10/5)^(2*5) +(10/10)^(2*10) = 1750 because positive divisors of 10 are 1, 2, 5, 10.

Seiichi Manyama, Table of n, a(n) for n = 1..3143

Cf. A038041, A055225, A151954, A236696.

Table[Total[Quotient[n, x = Divisors[n]]^(2*x)], {n, 34}] (* Jayanta Basu, Jul 08 2013 *)

a(n):= lsum(d^(2*n/d),d,listify(divisors(n)));
makelist(a(n),n,1,40); /* Emanuele Munarini , Feb 03 2014 */

a(n)=sumdiv(n, d, (d)^(2*n/d) );  /* Joerg Arndt, Oct 07 2012 */

G.f.: Sum_{n>=1} -log(1 - (n^2)*x^n)/n = Sum_{n>=1} a(n) x^n/n.

G.f.: Sum_{k>=1} k^2*x^k/(1-k^2*x^k). - Benoit Cloitre, Apr 21 2003

Corrected a(14) and inserted missing a(16) by Jayanta Basu, Jul 08 2013

A294620 Expansion of Product_{k>0} (1 - k^2*x^k)^(1/k).

Original entry on oeis.org

1, -1, -2, -1, -3, 7, -12, 32, -10, -15, 77, 187, -760, 846, 1382, -4197, 1371, 6650, -9991, 19220, -32439, -80889, 290596, 127853, -1372003, 913414, 3253746, -6728692, 2302327, 14461937, -46087740, 66588519, 15702643, -357119564, 752905288, 310992687

Offset: 0

Seiichi Manyama, Nov 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..3155

Cf. A073705, A151954.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k^2*x^k)^(1/k)))

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A073705(k)*a(n-k) for n > 0.

A303354 Expansion of Product_{k>0} (1+k^2*x^k)^(-1/k).

Original entry on oeis.org

1, -1, -1, -2, 4, -3, 3, -6, 75, -152, -4, -21, 1136, -1118, -1348, -5846, 20189, -11851, 17440, -49133, 119449, -448210, 72614, 35800, 6048178, -6376555, -5239650, -25634644, 52463464, -20483411, 155646867, -229344925, 664833536, -2469711562, 819287282

Offset: 0

Seiichi Manyama, Apr 22 2018

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This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/n, g(n) = -n^2.

Cf. A151954, A303355.

seq(coeff(series(mul((1+k^2*x^k)^(-1/k), k = 1..n), x, n+1), x, n), n = 0..40); # Muniru A Asiru, Apr 22 2018

cp's OEIS Frontend

A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

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A294620 Expansion of Product_{k>0} (1 - k^2*x^k)^(1/k).

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A303354 Expansion of Product_{k>0} (1+k^2*x^k)^(-1/k).

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