A308836 -id:A308836 - cp's OEIS frontend

A005797 Expansion of Jacobi nome q in terms of parameter m/16.

0, 1, 8, 84, 992, 12514, 164688, 2232200, 30920128, 435506703, 6215660600, 89668182220, 1305109502496, 19138260194422, 282441672732656, 4191287776164504, 62496081197436736, 935823746406530603, 14065763582458332888, 212122153814497767004, 3208590886304243284640

Offset: 0

N. J. A. Sloane

For a faster convergent series see A002103, where k' = sqrt(1 - k^2). - Wolfdieter Lang, Jul 14 2016

The Ansatz technique of A308835, A308836, and A308837 also works to produce the coefficients of this sequence from the ODE: T-d/dx(4*(1-x)*x*dT/dx)=0. - Bradley Klee, Jul 03 2019

			G.f. = x + 8*x^2 + 84*x^3 + 992*x^4 + 12514*x^5 + 164688*x^6 + 2232200*x^7 + ...
Given g.f. A(x),  then q = exp(-Pi sqrt(6)) = A( m/16 ) where m = ((2-sqrt(3))*(sqrt(3)-sqrt(2)))^2. - _Michael Somos_, Oct 30 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
C. L. Mallows, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..700
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
Index entries for reversions of series

Reversion of A005798. Cf. A002639. Other nomes: A308835, A308836, A308837.

a:= n-> coeff(series(EllipticNome(4*sqrt(x)), x, n+1), x, n):
seq(a(n), n=0..17);  # Thomas Richard, Aug 03 2022

a[ n_] := SeriesCoefficient[ EllipticNomeQ[ 16 x], {x, 0 ,n}] (* Michael Somos, Jul 11 2011 *)

{a(n) = if( n < 1, 0, polcoeff( serreverse( x * prod(k=1, n-1, (1 + x^k)^(-1)^k, 1 + x * O(x^n))^8), n))} /* Michael Somos, Jul 19 2002 */

{a(n) = my(A, m); if( n < 1, 0, m=1; A = x + O(x^2); while( m < n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff( serreverse(A), n))} /* Michael Somos, Mar 18 2003 */

G.f.: q = q(m) = Sum_{n>=0} a(n) * (m/16)^n.

G.f.: exp( -Pi * agm(1, sqrt(1 - 16 * x) ) / agm(1, sqrt( 16*x ) ) ).

A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/27)^n follows from the series solutions of 2T-d/dx(9(1-x)x*dT/dx)=0.

Original entry on oeis.org

0, 1, 15, 279, 5729, 124554, 2810718, 65114402, 1538182398, 36887880105, 895303119303, 21943398532563, 542209373589501, 13489931811324550, 337599511395854298, 8491805574767197650, 214548940430198454054, 5441921826542937659088, 138512110164878076019560

Offset: 0

Bradley Klee, Jun 27 2019

nonn

Also appears in Ramanujan's theory of elliptic functions, signature 3 (cf. A006480). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}.

B. C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.

G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275. [The article is on pages 273-295.]

Cf. A005797, A308836, A308837.

G[nMax_]:=Dot[RecurrenceTable[{Dot[{(3*n-5)^2 (3*n-4)^2 (9*n-4), -18(n - 1)(40 - 197*n + 351*n^2 - 279*n^3 + 81*n^4),81(n - 1)*n^3*(9*n - 13)}, a[n-#] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 5/9}, a, {n, 0, nMax}], x^Range[0, nMax]];
qSer[nMax_] := Expand[Times[x, Normal[Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/3, 2/3, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[20]/.{x->k*x},x]/.{k->27}

A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/432)^n follows from the series solutions of 5T-d/dx(36(1-x)x*dT/dx)=0.

Original entry on oeis.org

0, 1, 312, 107604, 39073568, 14645965026, 5609733423408, 2182717163349896, 859521859502348352, 341679883727799750159, 136868519056531319862408, 55173969942211048781835468, 22360181278518828446785034976, 9103073677708423854325869548662

Offset: 0

Bradley Klee, Jun 27 2019

nonn

Also appears in Ramanujan's theory of elliptic functions, signature 6 (cf. A113424). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}.

B.C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.

G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275.

Cf. A005797, A308835, A308836.

G[nMax_] := Dot[RecurrenceTable[{Dot[{(6*n - 11)^2 (6*n - 7)^2 (18*n - 5), -36 (n - 1) (385 - 2426*n + 4968*n^2 - 4248*n^3 + 1296*n^4), 1296 (n - 1) n^3 (18*n - 23)},
a[n - #] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 13/18}, a, {n, 0, nMax}], x^Range[0, nMax]];
qSer[nMax_] := Expand[Times[x,Normal[ Series[Exp[Divide[G[nMax], Hypergeometric2F1[1/6, 5/6, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[12] /. {x -> k*x}, x] /. {k -> 432}

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A005797 Expansion of Jacobi nome q in terms of parameter m/16.

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A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/27)^n follows from the series solutions of 2T-d/dx(9(1-x)x*dT/dx)=0.

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A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/432)^n follows from the series solutions of 5T-d/dx(36(1-x)x*dT/dx)=0.

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cp's OEIS Frontend

A005797 Expansion of Jacobi nome q in terms of parameter m/16.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/27)^n follows from the series solutions of 2*T-d/dx(9*(1-x)*x*dT/dx)=0.

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A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/432)^n follows from the series solutions of 5*T-d/dx(36*(1-x)*x*dT/dx)=0.

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Author

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Mathematica

A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/27)^n follows from the series solutions of 2T-d/dx(9(1-x)x*dT/dx)=0.

A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/432)^n follows from the series solutions of 5T-d/dx(36(1-x)x*dT/dx)=0.