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A053992
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The number phi_3(n) of Frobenius partitions that allow up to 3 repetitions of an integer in a row.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| Andrews, George E., Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.
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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)]
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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 + ...
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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 */
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CROSSREFS
| Sequence in context: A147079 A152074 A123629 * A052825 A003082 A058053
Adjacent sequences: A053989 A053990 A053991 * A053993 A053994 A053995
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KEYWORD
| easy,nonn
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AUTHOR
| James A. Sellers (sellersj(AT)math.psu.edu), Apr 04 2000
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