A235324 Sum of all parts of all partitions of n into an even number of parts minus the sum of all parts of all partitions of n into an odd number of parts.

-1, 0, -3, 4, -5, 6, -7, 16, -18, 20, -22, 36, -39, 42, -60, 80, -85, 90, -114, 140, -168, 176, -207, 264, -300, 312, -378, 448, -493, 540, -620, 736, -825, 884, -1015, 1188, -1295, 1406, -1599, 1840, -2009, 2184, -2451, 2772, -3060, 3312, -3666, 4176, -4557

Offset

1,3

Comments

Zero together with the absolute values of this sequence give A330373. - Omar E. Pol, Dec 31 2019

Links

Clark Kimberling, Table of n, a(n) for n = 1..500

Formula

a(n) = n*A081362(n) for n >= 1.

G.f.: x*f'(x), where f(x) = Product_{k > 0} (1 - x^(2*k - 1)).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) * n^(1/4) / (2^(7/4) * 3^(1/4)). - Vaclav Kotesovec, Nov 21 2016

Example

There are 12 partitions of 8 into an even number of parts and 10 into an odd number of parts, so that a(8) = 12*8 - 10*8 = 16.

Mathematica

Table[n*SeriesCoefficient[Product[1-x^k, {k, 1, n, 2}], {x, 0, n}], {n, 70}]
nmax = 60; Rest[CoefficientList[Series[x*D[Product[1/(1 + x^k), {k, 1, nmax}], x], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 21 2016 *)

Prog

(PARI) a(n)= my(A); n*if(n<0, 0, A= x*O(x^n); polcoeff(eta(x+A)/eta(x^2+A), n));
vector(50, n, a(n) ) /* from Michael Somos code in A081362 */ \\ G. C. Greubel, Dec 31 2019

Crossrefs

Cf. A000041, A027187, A027193, A081362, A330373.

Sequence in context: A361267 A070916 A078305 * A330373 A072599 A095138

Adjacent sequences: A235321 A235322 A235323 * A235325 A235326 A235327

Keyword

easy,sign

Author

Clark Kimberling, Mar 27 2014

Status

approved