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Search: id:A109368
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| A109368 |
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Partitions of n into parts relatively prime to 42. |
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2
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| 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 27, 30, 34, 37, 40, 44, 49, 54, 60, 65, 71, 78, 85, 94, 103, 112, 122, 132, 144, 158, 172, 186, 201, 218, 237, 258, 279, 302, 326, 352, 381, 412, 445, 480, 516, 556, 599, 646
(list; graph; listen)
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OFFSET
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0,6
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FORMULA
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Euler transform of period 42 sequence [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, ...].
G.f.: Product_{k>0} (1+x^k)(1+x^(21k))/((1+x^(3k))(1+x^(7k))) = Product_{k>0} P42(x^k) where P42 is the 42nd cyclotomic polynomial.
Expansion of chi(-q^3) * chi(-q^7) / (chi(-q) * chi(-q^21)) in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(-1/2) * eta(q^2) * eta(q^3) * eta(q^7) * eta(q^42) / (eta(q) * eta(q^6) * eta(q^14) * eta(q^21)) in powers of q.
Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (1 + v) * (v^2 - u^2*w^2) - (v - v^2) * (u^2 + w^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (168 t)) = 1 / f(t) where q = exp(2 pi i t).
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EXAMPLE
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q + q^3 + q^5 + q^7 + q^9 + 2*q^11 + 2*q^13 + 2*q^15 + 2*q^17 + 2*q^19 + ...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)*eta(x^7+A)*eta(x^42+A)/ (eta(x+A)*eta(x^6+A)*eta(x^14+A)*eta(x^21+A)), n))}
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CROSSREFS
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Convolution inverse of A112211.
Sequence in context: A109697 A103373 A038539 this_sequence A046774 A029105 A079954
Adjacent sequences: A109365 A109366 A109367 this_sequence A109369 A109370 A109371
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Jun 26 2005, Jan 12 2009
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