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A213598
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Number of partitions of n in which no parts are multiples of 49.
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3
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1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173524
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OFFSET
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0,3
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COMMENTS
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In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).
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REFERENCES
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R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)
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LINKS
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FORMULA
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Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.
Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].
Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),
B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).
a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 13 2014 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};
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CROSSREFS
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Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A035959 (m=5), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10), A092885 (m=25), this sequence (m=49).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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