A187053 -id:A187053 - cp's OEIS frontend

A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

1, -1, 2, -3, 4, -6, 9, -12, 16, -22, 29, -38, 50, -64, 82, -105, 132, -166, 208, -258, 320, -395, 484, -592, 722, -876, 1060, -1280, 1539, -1846, 2210, -2636, 3138, -3728, 4416, -5222, 6163, -7256, 8528, -10006, 11716, -13696, 15986, -18624, 21666, -25169, 29190, -33808, 39104

Offset: 0

Michael Somos, Apr 24 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A079006.
Convolution inverse is A029838.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 6*x^5 + 9*x^6 - 12*x^7 + 16*x^8 - 22*x^9 + ...
G.f. = q - q^9 + 2*q^17 - 3*q^25 + 4*q^33 - 6*q^41 + 9*q^49 - 12*q^57 + 16*q^65 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
H. T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York 1962, p. 170 MR0181773 (31 #6000)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A001935, A029838, A108494.

(psi(x) / phi(x))^b: this sequence (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), A320050 (b=7).

phi[x_] := EllipticTheta[3, 0, x]; psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)]; s = Series[ psi[x]/phi[x], {x, 0, 100}]; A083365 = CoefficientList[s, x] (* Jean-François Alcover, Feb 18 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^2/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
(QPochhammer[-x^2, x^2, -1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 10 2019~ *)

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x * A^2))); polcoeff(sqrt(A), n))};

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[2,1] / A[1,1] + x * O(x^n), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3, n))};

Expansion of f(-x^4) / f(x) = psi(x) / phi(x) = psi(x^2) / psi(x) = psi(-x) / phi(-x^2) = 1 / (chi(x) * chi(-x^2)) = 1 / (chi^2(x) * chi(-x)) = chi(-x) / chi^2(-x^2) = (psi(x^2) / phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of k^(1/4) / (2^(1/2) * q^(1/8)) in powers of q where k is elliptic modulus and q is the nome.
Expansion of q^(-1/8) * eta(q) * eta(q^4)^2 / eta(q^2)^3 in powers of q.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^4 * (1 + 4*v^4).
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = v^4 - u^4 + u*v + 4*(u*v)^3.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w - u^2*v*(1 + 2*w^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u6 - u1*u3 * (u2^2 + u6^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = sqrt(2) * q * A(q^8) satisfies 0 = f(B(q), B(q^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
Euler transform of period 4 sequence [-1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A108494. - Michael Somos, Feb 29 2012
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)) = (Sum_{k>0} x^(k^2 - k)) / (Sum_{k>0} x^((k^2 - k)/2)).
G.f.: 1 / (1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...)))).
A001935(n) = (-1)^n a(n).
G.f.: (1+1/Q(0))/2, where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2))/(2^(11/4)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
G.f.: (-x^2; x^2){-1/2} = ((-1; x^2){1/2})/2, where (a; q)n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(11/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k))). - Ilya Gutkovskiy, May 28 2018

A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 9, 22, 48, 99, 194, 363, 657, 1155, 1977, 3312, 5443, 8787, 13968, 21894, 33873, 51795, 78345, 117312, 174033, 255945, 373353, 540486, 776848, 1109040, 1573209, 2218198, 3109713, 4335840, 6014123, 8300811, 11402928, 15593702, 21232521, 28790667, 38884082

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

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

The Cayley reference is actually to A187053. - Michael Somos, Jul 26 2012

			1 + 3*x + 9*x^2 + 22*x^3 + 48*x^4 + 99*x^5 + 194*x^6 + 363*x^7 + 657*x^8 + ...
q^3 + 3*q^11 + 9*q^19 + 22*q^27 + 48*q^35 + 99*q^43 + 194*q^51 + 363*q^59 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941.

g100:= mul((1+x^(2*k))/(1-x^(2*k-1)),k=1..50)^3:
S:= series(g100,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Nov 30 2015

CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] (* Olivier GERARD, May 06 2009 *)
QP = QPochhammer; s = (QP[q^4]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^3, n))} /* Michael Somos, Mar 06 2011 */

Expansion of q^(-3/8) * (eta(q^4) / eta(q))^3 in powers of q. - Michael Somos, Jul 26 2012
Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - Michael Somos, Mar 06 2011
Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - Michael Somos, Mar 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.
a(n) ~ 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Corrected and extended by Simon Plouffe

Checked and more terms from Olivier GERARD, May 06 2009

A195861 Expansion of (psi(x) / phi(x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 20, -65, 185, -481, 1165, -2665, 5820, -12220, 24802, -48880, 93865, -176125, 323685, -583798, 1035060, -1806600, 3108085, -5276305, 8846884, -14663645, 24044285, -39029560, 62755345, -100004806, 158022900, -247710570, 385366265

Offset: 0

Michael Somos, Sep 24 2011

sign

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

			G.f. = 1 - 5*x + 20*x^2 - 65*x^3 + 185*x^4 - 481*x^5 + 1165*x^6 - 2665*x^7 + ...
G.f. = q^5 - 5*q^13 + 20*q^21 - 65*q^29 + 185*q^37 - 481*q^45 + 1165*q^53 - 2665*q^61 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001939, A029842.

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), this sequence (b=5).

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16)^(5/8), {q, 0, n + 5/8}]];
a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^5, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2])^5, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^2)^5, n))};

Expansion of q^(-5/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^5 in powers of q.
Euler transform of period 4 sequence [-5, 10, -5, 0, ...].
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^5.
a(n) = (-1)^n * A001939(n). Convolution inverse of A029842.
a(n) ~ (-1)^n * 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 27 2015

A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -6, 27, -98, 309, -882, 2330, -5784, 13644, -30826, 67107, -141444, 289746, -578646, 1129527, -2159774, 4052721, -7474806, 13569463, -24274716, 42838245, -74644794, 128533884, -218881098, 368859591, -615513678, 1017596115, -1667593666, 2710062756, -4369417452

Offset: 0

Seiichi Manyama, Oct 04 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), this sequence (b=6), A320050 (b=7).

Cf. A029843.

nmax = 40; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Convolution inverse of A029843.
Expansion of q^(-3/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^6 in powers of q.
a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n)) / (128*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -7, 35, -140, 483, -1498, 4277, -11425, 28889, -69734, 161735, -362271, 786877, -1662927, 3428770, -6913760, 13660346, -26492361, 50504755, -94766875, 175221109, -319564227, 575387295, -1023624280, 1800577849, -3133695747, 5399228149, -9214458260, 15584195428

Offset: 0

Seiichi Manyama, Oct 04 2018

sign

In general, for b > 0 and (psi(x) / phi(x))^b, a(n) ~ (-1)^n * b^(1/4) * exp(Pi*sqrt(b*(n/2))) / (2^(b + 7/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), this sequence (b=7).

Cf. A029844.

nmax = 50; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Convolution inverse of A029844.
Expansion of q^(-7/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^7 in powers of q.
a(n) ~ (-1)^n * 7^(1/4) * exp(Pi*sqrt((7*n)/2)) / (256*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

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A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

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A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.

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A195861 Expansion of (psi(x) / phi(x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

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A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.

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A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.

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