A053992 The number phi_3(n) of Frobenius partitions that allow up to 3 repetitions of an integer in a row.

1, 1, 3, 6, 11, 18, 31, 49, 78, 119, 180, 267, 394, 567, 813, 1151, 1616, 2244, 3099, 4240, 5769, 7790, 10462, 13965, 18552, 24502, 32223, 42176, 54972, 71340, 92242, 118800, 152481, 195017, 248621, 315945, 400315, 505694, 637068, 800380, 1002964

Offset

0,3

References

Andrews, George E., Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984. See phi_3(n), page 41.

Links

Table of n, a(n) for n=0..40.

Formula

Expansion of q^(1/8) * eta(q^6)^5 / (eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q. - Michael Somos, Mar 09 2011

Euler transform of period 12 sequence [ 1, 2, 3, 2, 1, -1, 1, 2, 3, 2, 1, 1, ...]. - Michael Somos, Mar 09 2011

G.f.: Product_{k>0} (1 - x^(12*k - 6)) / ( (1 - x^(6*k - 5)) * (1 - x^(6*k - 4))^2 * (1 - x^(6*k - 3))^3 * (1 - x^(6*k - 2))^2 * (1 - x^(6*k - 1)) *(1 - x^(12*k)) ). [Andrews, p. 11, equation (5.10)]

a(n) ~ exp(sqrt(n)*Pi)/(4*sqrt(6)*n). - Vaclav Kotesovec, Nov 13 2016

Example

1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 31*x^6 + 49*x^7 + 78*x^8 + ...
1/q + q^7 + 3*q^15 + 6*q^23 + 11*q^31 + 18*q^39 + 31*q^47 + 49*q^55 + 78*q^63 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k - 3)) / ( (1 - x^(6*k - 5)) * (1 - x^(6*k - 4))^2 * (1 - x^(6*k - 3))^2 * (1 - x^(6*k - 2))^2 * (1 - x^(6*k - 1)) * (1 - x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))} /* Michael Somos, Mar 09 2011 */

Crossrefs

Sequence in context: A212484 A279100 A347415 * A052825 A003082 A058053

Adjacent sequences: A053989 A053990 A053991 * A053993 A053994 A053995

Keyword

easy,nonn

Author

James A. Sellers, Apr 04 2000

Status

approved