A195130 -id:A195130 - cp's OEIS frontend

A005149 Sequence of coefficients arising in connection with a rapidly converging series for Pi.

1, -24, 852, -35744, 1645794, -80415216, 4094489992, -214888573248, 11542515402255, -631467591949480, 35063515239394764, -1971043639046131296, 111949770626330347638, -6414671157989386260432, 370360217892318010055832, -21525284426246779936288192

Offset: 1

Simon Plouffe and N. J. A. Sloane

			x - 24*x^2 + 852*x^3 - 35744*x^4 + 1645794*x^5 - 80415216*x^6 + 4094489992*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 1..100
M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217.
Index entries for sequences related to the number Pi
Index entries for reversions of series

Cf. A005148, A014103, A195130.

CoefficientList[InverseSeries[Series[x*Product[(1+x^k)^24, {k, 1, 16}], {x, 0, 16}], x], x] // Rest (* Jean-François Alcover, Mar 29 2011 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ q Product[ 1 + q^k, {k, n}]^24, {q, 0, n}], x], {x, 0, n}]] (* Michael Somos, Sep 11 2011 *)

{a(n) = if( n<1, 0, polcoeff( serreverse( x * prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^24), n))}

REVERT(A014103). a(n) = -(-1)^n * A195130(n).

Formula and more terms from Michael Somos, Nov 24 2001

Signs corrected, Dec 24 2001

A100130 Expansion of (eta(q) * eta(q^4) / eta(q^2)^2)^24 in powers of q.

Original entry on oeis.org

1, -24, 300, -2624, 18126, -105504, 538296, -2471424, 10400997, -40674128, 149343012, -519045888, 1718732998, -5451292992, 16633756008, -49010118656, 139877936370, -387749049720, 1046413709980, -2754808758144, 7087483527072

Offset: 1

Michael Somos, Nov 06 2004

sign

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

			G.f. = q - 24*q^2 + 300*q^3 - 2624*q^4 + 18126*q^5 - 105504*q^6 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A014103, A097340, A195130.

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) m / 16, {q, 0, n}]];
a[ n_] := SeriesCoefficient[ q / Product[ 1 + q^k, {k, 1, n, 2}]^24, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q / QPochhammer[ -q, q^2]^24, {q, 0, n}];

{a(n) = polcoeff( x * prod(k=1, n, 1 + (-x)^k, 1 + x * O(x^n))^24, n)};

{a(n) = my(A); if( n<1, 0,  n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^24, n))};

Expansion of q / chi(q)^24 in powers of q where chi() is a Ramanujan theta function.
Expansion of lambda * (1 - lambda) / 16 in powers of q.
Euler transform of period 4 sequence [ -24, 24, -24, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 4096 * (u*v)^4 + (u*v)^2 * (1791 + 2352 * (u + v) - 10496 * u*v) - u*v * (1 - 48 * (u + v) + 96 * (u^2 + v^2)) + u^3 + v^3.
G.f.: x * (Product_{k>0} (1 + (-x)^k))^24 = x / (Product_{k>0} (1 + x^(2*k - 1)))^24.
a(n) = -(-1)^n * A014103(n). Convolution inverse of A097340. Series reversion of A195130.
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(2*n)) / (4096 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018

Swapped a formula with definition to make this clearer. - N. J. A. Sloane, Nov 26 2018

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A005149 Sequence of coefficients arising in connection with a rapidly converging series for Pi.

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A100130 Expansion of (eta(q) * eta(q^4) / eta(q^2)^2)^24 in powers of q.

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