A307311 - OEIS

%I #15 Sep 08 2022 08:46:21

%S 1,-1,-1,-2,-3,-4,-5,-6,-6,-1,19,74,200,461,977,1987,3976,7902,15559,

%T 30105,56778,103833,183765,314882,523007,841752,1305431,1916607,

%U 2540433,2609983,381628,-8814988,-36463325,-109113400,-285322360,-689608522,-1579574566,-3477967848

%N Expansion of Product_{k>=1} 1/(1 + x^k/(1 - x)^k).

%C First differences of the binomial transform of A081362.

%C Convolution inverse of A129519.

%H Robert Israel, <a href="/A307311/b307311.txt">Table of n, a(n) for n = 0..1000</a>

%p a:=series(mul(1/(1+x^k/(1-x)^k),k=1..100), x=0,38): seq(coeff(a, x, n), n=0..37); # _Paolo P. Lava_, Apr 02 2019

%t nmax = 40; CoefficientList[Series[Product[1/(1 + x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

%o (PARI) m=40; my(x='x+O('x^m)); Vec( 1/prod(k=1,m+2, (1+x^k/(1-x)^k)) ) \\ _G. C. Greubel_, Apr 03 2019

%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&*[(1+x^k/(1-x)^k): k in [1..m+2]]) )); // _G. C. Greubel_, Apr 03 2019

%o (Sage) m=40; (1/product(1+x^k/(1-x)^k for k in (1..m+2))).series(x, m).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 03 2019

%Y Cf. A081362, A129519, A218482, A307310.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Apr 02 2019