A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

0, 1, 0, 2, 2, 3, 3, 4, 7, 8, 9, 10, 15, 16, 18, 23, 30, 32, 35, 42, 51, 59, 63, 73, 89, 100, 106, 125, 145, 160, 174, 198, 229, 255, 274, 310, 355, 388, 420, 472, 534, 582, 631, 701, 784, 859, 928, 1021, 1144, 1243, 1338, 1475, 1630, 1767, 1909, 2089, 2299

Offset

0,4

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Formula

G.f.: A(q) = Sum_{n >= 1} n*q^(2*n-1)*(1+q)*(1+q^3)*...*(1+q^(2*n-3)).

From Peter Bala, Aug 20 2017: (Start)

Let F(q) = Product_{i >= 1} (1 + q^(2*i-1)). Then A(q) = Sum_{n >= 0} ( F(q) - Product_{i = 1..n} (1 + q^(2*i-1)) ).

It follows that the above sum A(q) satisfies -A(q-1) = 1 + q + 3*q^2 + 12*q^3 + 61*q^4 + ..., the g.f. for A158691, row-Fishburn matrices of size n. (End)

Mathematica

CoefficientList[Series[Sum[n*q^(2n-1)*Product[1+q^k, {k, 1, 2n-3, 2}], {n, 1, 30}], {q, 0, 60}], q]

Crossrefs

Cf. A000700, A000701, A006128, A015723, A046682, A079499, A158691.

Sequence in context: A367843 A164529 A153906 * A236972 A184351 A343659

Adjacent sequences: A067616 A067617 A067618 * A067620 A067621 A067622

Keyword

easy,nonn

Author

Naohiro Nomoto, Feb 01 2002

Extensions

Edited by Dean Hickerson, Feb 11 2002

Status

approved