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A053993
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The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.
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0
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1, 1, 3, 5, 9, 14, 24, 35, 55, 81, 120, 171, 248, 345, 486, 669, 920, 1246, 1690, 2256, 3014, 3984, 5253, 6870, 8970, 11618, 15022, 19306, 24745, 31557, 40154, 50845, 64244, 80850, 101501, 126982, 158514, 197218, 244865, 303143, 374497, 461435
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 16 2005
The sequence A077285 is the same but with multiplicities of all parts.
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REFERENCES
| Andrews, George E., Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.
Drake, Brian, Limits of areas under lattice paths. Discrete Math. 309 (2009), no. 12, 3936-3953.
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FORMULA
| Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos Mar 09 2011
Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - Michael Somos Mar 09 2011
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]
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EXAMPLE
| 1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - Michael Somos Mar 09 2011
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* Michael Somos Mar 09 2011 */
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CROSSREFS
| Cf. A077285.
Sequence in context: A120452 A144116 A061556 * A071155 A120695 A103578
Adjacent sequences: A053990 A053991 A053992 * A053994 A053995 A053996
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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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