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A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A241152 A164529 A153906 * A236972 A184351 A146922

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

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Last modified September 21 02:47 EDT 2017. Contains 292293 sequences.