A131945 Number of partitions of n where odd parts are distinct or repeated once.
1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 45, 55, 74, 90, 119, 145, 188, 228, 291, 351, 442, 532, 664, 796, 982, 1172, 1435, 1708, 2076, 2462, 2972, 3512, 4214, 4966, 5929, 6965, 8272, 9688, 11457, 13383, 15762, 18362, 21543, 25031, 29264, 33922, 39533, 45717
Offset: 0
Examples
a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1. G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ... G.f. = 1/q + q^5 + 2*q^11 + 2*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 10*q^41 + ...
Links
- Brian Drake and Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
- Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
- Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See p. 11.
- James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 2.
- Michael Somos, Introduction to Ramanujan theta functions.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Maple
A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20),q,21): seq(coeff(A,q,i), i=0..20);
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *) nmax = 50; CoefficientList[Series[Product[1 / ((1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 11 2016 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Aug 05 2007 */
Formula
G.f.: product_{n=1..oo} (1-q^(6n-3))/(1-q^n).
Expansion of chi(-x^3) / f(-x) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Aug 05 2007
Expansion of q^(1/6) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Aug 05 2007
Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - Michael Somos, Aug 05 2007
a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*n). - Vaclav Kotesovec, Dec 11 2016
Comments