A244465 -id:A244465 - cp's OEIS frontend

A244526 Expansion of f(-x^3, -x^5)^2 in powers of x where f() is Ramanujan's two-variable theta function.

1, 0, 0, -2, 0, -2, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, -2, 2, -2, 0, -2, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 2, -2, 0, 0, 3, 0, 2, -2, 0, 0, 2, 0, 2, 0, 0, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0, 0, -2, 0, -2, 1, 0, 2, 0, 0, -2, 2, -2, 2, 0, 0, 0, 3, 0, 0

Offset: 0

Michael Somos, Jun 29 2014

sign

			G.f. = 1 - 2*x^3 - 2*x^5 + x^6 + 2*x^8 + x^10 + 2*x^14 - 2*x^17 + 2*x^18 + ...
G.f. = q - 2*q^25 - 2*q^41 + q^49 + 2*q^65 + q^81 + 2*q^113 - 2*q^137 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A244465.

 QP := QPochhammer; A244526[n_]:= SeriesCoefficient[(QP[q^3, q^8]*QP[q^5, q^8]*QP[q^8])^2, {q, 0, n}]; Table[A244526[n], {n, 0, 50}] (* G. C. Greubel, Dec 25 2017 *)

{a(n) = (-1)^n * sum(k=0, n, issquare(16*k + 1) * issquare(16*(n-k) + 1))};

Euler transform of period 8 sequence [ 0, 0, -2, 0, -2, 0, 0, -2, ...].
G.f.: f(-x^3, -x^5)^2 = (Sum_{k in Z} (-1)^k * x^(4*k^2 - k))^2.
Convolution square of A244465.
a(9*n + 4) = a(9*n + 7) = 0. a(49*n + 6) = a(n).

A143433 Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Aug 14 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x + x^3 - x^6 - x^10 + x^15 - x^21 + x^28 + x^36 - x^45 + x^55 + ...
G.f. = q - q^9 + q^25 - q^49 - q^81 + q^121 - q^169 + q^225 + q^289 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A143434, A244465, A244525.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^4] QPochhammer[ -x^3, -x^4] QPochhammer[ -x^4], {x, 0, n}]; (* Michael Somos, Jun 03 2015 *)

{a(n) = if( n<0, 0, if( issquare(8*n + 1, &n), n = n\2; (-1)^(n + n\4), 0))};

{a(n) = my(A); if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^( [1, 1, 0, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, 0, 1, 1] [k%16 + 1]), 1 + x * O(x^n)), n))};

Euler transform of period 16 sequence [ -1, 0, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 0, -1, -1, ...].
Pattern of signs of nonzero terms is A143431.
G.f.: Sum_{k>=0} (-1)^(k + floor(k/4)) * x^(k * (k+1) / 2).
a(n) = (-1)^n * A143434(n).
a(2*n) = A244465(n). a(2*n + 1) = - A244525(n). a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = 0.

A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k(4k + 1)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 8, 4, 9, 9, 6, 15, 10, 12, 20, 12, 22, 23, 18, 35, 26, 30, 46, 32, 51, 54, 45, 76, 62, 71, 99, 76, 111, 117, 104, 160, 136, 154, 205, 167, 230, 244, 223, 319, 286, 319, 406, 349, 456, 484, 458, 619, 570, 632, 779, 695

Offset: 0

Ilya Gutkovskiy, May 24 2019

nonn

Number of partitions of n into parts congruent to {0, 3, 5} mod 8.

Convolution inverse of A244465.

			For n=23 the a(23)=6 solutions are 3+3+3+3+3+3+5, 3+3+3+3+3+8, 3+3+3+3+11, 3+5+5+5+5, 5+5+5+8, and 5+5+13.

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000009, A000041, A006950, A007742, A047622, A074378, A195850, A244465, A308400.

nmax = 68; CoefficientList[Series[1/Sum[(-x)^(k (4 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Product[1/((1 - x^(8 k - 5)) (1 - x^(8 k - 3)) (1 - x^(8 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Sum[PartitionsP[k] (-x)^k, {k, 0, nmax}]/Sum[PartitionsQ[2 k] (-x)^k, {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: 1 / Sum_{k>=0} (-x)^A074378(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(8*k - 5)) * (1 - x^(8*k - 3)) * (1 - x^(8*k))).
G.f.: ( Sum_{k>=0} A000041(k)*(-x)^k ) / ( Sum_{k>=0} A000009(2*k)*(-x)^k ).
a(n) ~ sqrt(sqrt(2) - 1) * exp(sqrt(n)*Pi/2) / (2^(9/4)*n). - Vaclav Kotesovec, May 25 2019
a(n) = a(n-3) + a(n-5) - a(n-14) - a(n-18) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 5, 14, 18, ... is the sequence A074378. - Ludovic Schwob, Aug 04 2021

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A244526 Expansion of f(-x^3, -x^5)^2 in powers of x where f() is Ramanujan's two-variable theta function.

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A143433 Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general theta function.

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A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k(4k + 1)).

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cp's OEIS Frontend

A244526 Expansion of f(-x^3, -x^5)^2 in powers of x where f() is Ramanujan's two-variable theta function.

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A143433 Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general theta function.

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PARI

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A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(4*k + 1)).

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A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k(4k + 1)).