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A098098
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a(n) = sigma(6*n+5)/6.
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16
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1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72
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OFFSET
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0,2
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COMMENTS
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Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020
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LINKS
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Ivan Neretin, Table of n, a(n) for n = 0..10000
Nayandeep Deka Baruah and Kallol Nath, Infinite families of arithmetic identities and congruences for bipartitions with 3-cores, Journal of Number Theory, Volume 149, April 2015, Pages 92-104.
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FORMULA
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G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) = A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
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EXAMPLE
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G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...
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MATHEMATICA
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Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)
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PROG
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(PARI) a(n) = sigma(6*n + 5)/6
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */
(MAGMA) Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */
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CROSSREFS
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Cf. A000203, A033686, A097723, A121443, A125514, A133739, A134077, A134078, A134079, A186100, A232343, A232356, A252650, A238831.
Sequence in context: A332816 A094607 A258831 * A326066 A080785 A319605
Adjacent sequences: A098095 A098096 A098097 * A098099 A098100 A098101
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KEYWORD
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easy,nonn
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AUTHOR
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Vladeta Jovovic, Sep 14 2004
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STATUS
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approved
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