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A098613
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Expansion of psi(x^2) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
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7
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1, 1, 3, 4, 7, 10, 17, 23, 35, 48, 69, 93, 131, 173, 236, 310, 413, 536, 704, 903, 1170, 1489, 1904, 2403, 3044, 3811, 4784, 5951, 7409, 9157, 11325, 13912, 17095, 20891, 25519, 31029, 37708, 45632, 55184, 66495, 80050, 96064, 115173, 137680, 164425, 195860
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of partitions of n in which each odd part can occur any number of times but each even part is of two kinds and each kind can occur at most once. - Michael Somos, Dec 01 2019
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LINKS
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FORMULA
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Expansion of chi(x) / chi(-x^2)^3 = 1 / (chi(-x)* chi(-x^2)^2) = 1 / (chi(x)^2 * chi(-x)^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 07 2015
Expansion of q^(-5/24) * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [1, 2, 1, 0, ...].
G.f. A(x) is the limit of x^(n^2+n) * P_{2*n+1}(1/x)/2 where P_n(q) = Sum_{k=0..n} C(n, k; q) and C(n, k; q) is the q-binomial coefficients. See A083906 for P_n(q).
G.f.: (Sum_{k>0} x^(k^2-k)) / (Product_{k>0} (1 - x^k)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143161. - Michael Somos, Sep 07 2015
G.f.: Product_{k>=1} (1 + x^(2*k))^2 / (1 - x^(2*k-1)). - Michael Somos, Dec 01 2019
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EXAMPLE
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G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 23*x^7 + ...
G.f. = q^5 + q^29 + 3*q^53 + 4*q^77 + 7*q^101 + 10*q^125 + 17*q^149 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
nmax = 40; CoefficientList[ Series[Product[(1 + x^k) * (1 + x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^2]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x]^2 QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)) / eta(x + x * O(x^n)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, 2*n+1, prod(i=1, k, (1 - x^(2*n+2 - i)) / (1 - x^i))) / 2, n^2))};
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CROSSREFS
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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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