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A107742 G.f.: prod(j>=1, prod(i>=1, 1+x^(i*j) ) ). 37
1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Transforms

FORMULA

Euler transform of A001227.

Weigh transform of A000005.

G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). [Paul D. Hanna, Jun 26 2005]

G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). [Paul D. Hanna, Mar 28 2009]

G.f.: prod(n>=1, Q(x^n) ) where Q(x) is the g.f. of A000009. [Joerg Arndt, Feb 27 2014]

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017

Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)

nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)

nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, 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}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)

PROG

(PARI) a(n)=polcoeff(prod(k=1, n, prod(j=1, n\k, 1+x^(j*k)+x*O(x^n))), n) /* Paul D. Hanna */

(PARI) N=66;  x='x+O('x^N); gf=1/prod(j=0, N, eta(x^(2*j+1))); gf=prod(j=1, N, (1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */

(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)), n))} /* Paul D. Hanna, Mar 28 2009 */

CROSSREFS

Cf. A006171, A109386, A219554, A280473, A280486, A288007.

Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

Sequence in context: A204656 A070689 A091611 * A228779 A158510 A004695

Adjacent sequences:  A107739 A107740 A107741 * A107743 A107744 A107745

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Jun 11 2005

EXTENSIONS

More terms from Paul D. Hanna, Jun 26 2005

STATUS

approved

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)