OFFSET
6,2
COMMENTS
Number of partitions of n with three designated summands. For example: a(8) = 9 because there are 9 partitions of 8 with three designated summands: [5'+ 2'+ 1'], [4'+ 3'+ 1'], [4'+ 2'+ 1'+ 1], [4'+ 2'+ 1 + 1'], [3'+ 2'+ 2 + 1'], [3'+ 2 + 2'+ 1'], [3'+ 2'+ 1'+ 1 + 1], [3'+ 2'+ 1 + 1'+ 1], [3'+ 2'+ 1 + 1 + 1']. - Omar E. Pol, Jul 25 2025
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
John Cerkan, Table of n, a(n) for n = 6..10000
G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
FORMULA
G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum(x^(n*i)/(1-x^n)^(2*i),n=1..inf), i=1..3. - Vladeta Jovovic, Sep 21 2007
G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+3, 6) * x^( k*(k+1) / 2 )) / (-7 * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - Michael Somos, Jan 10 2012
Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / (6!*7!). - Vaclav Kotesovec, Aug 01 2025
EXAMPLE
x^6 + 3*x^7 + 9*x^8 + 22*x^9 + 42*x^10 + 81*x^11 + 140*x^12 + 231*x^13 + ...
PROG
(PARI) {a(n) = if( n<1, 0, (3*sigma(n, 5) + (-30*n + 50)*sigma(n, 3) + (40*n^2 - 100*n + 37)*sigma(n)) / 1920)} /* Michael Somos, Jan 10 2012 */
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms from Naohiro Nomoto, Jan 24 2002
More terms from Vladeta Jovovic, Sep 21 2007
STATUS
approved