A001940 -id:A001940 - cp's OEIS frontend

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

1, 1, 5, 22, 101, 481, 2330, 11425, 56549, 281911, 1413465, 7120136, 36006362, 182681916, 929461993, 4740491107, 24229115109, 124069449335, 636376573943, 3268955179686, 16814509004601, 86593280920756, 446437797872016, 2303948443259841, 11900990745759578, 61526182236027756

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Main diagonal of A296068.

Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.

Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 - x^(4*k))/(1 - x^k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - Vaclav Kotesovec, Dec 05 2017

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

A001941 Absolute values of coefficients of an elliptic function.

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

N. J. A. Sloane, Simon Plouffe

nonn

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).

T. D. Noe, Table of n, a(n) for n = 0..1000
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]

Cf. A001935, A001936, A001937, A001939, A001940, A092877, A093160.

nn = 4*10; b = Flatten[Table[{7, 7, 7, 0}, {nn/4}]]; CoefficientList[x*Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
nmax = 40; CoefficientList[Series[Product[((1 - x^(4*k)) / (1 - x^k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2017 *)

G.f.: Product ( 1 - x^k )^-c(k), c(k) = 7, 7, 7, 0, 7, 7, 7, 0, ....
a(n) ~ 7^(1/4) * exp(sqrt(7*n/2)*Pi) / (256*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^7. - Ilya Gutkovskiy, Dec 04 2017

A296068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j))/(1 - x^(2j-1)))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 4, 0, 1, 5, 14, 22, 18, 6, 0, 1, 6, 20, 40, 48, 32, 9, 0, 1, 7, 27, 65, 101, 99, 55, 12, 0, 1, 8, 35, 98, 185, 236, 194, 90, 16, 0, 1, 9, 44, 140, 309, 481, 518, 363, 144, 22, 0, 1, 10, 54, 192, 483, 882, 1165, 1080, 657, 226, 29, 0, 1, 11, 65, 255, 718, 1498, 2330, 2665, 2162, 1155, 346, 38, 0

Offset: 0

Ilya Gutkovskiy, Dec 04 2017

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 3)*x^2 + (1/6)*k*(k^2 + 9*k + 8)*x^3 + (1/24)*k*(k^3 + 18*k^2 + 59*k + 18)*x^4 + (1/120)*k*(k^4 + 30*k^3 + 215*k^2 + 330*k + 144)*x^5 + ...
Square array begins:
1,  1,   1,   1,    1,    1,  ...
0,  1,   2,   3,    4,    5,  ...
0,  2,   5,   9,   14,   20,  ...
0,  3,  10,  22,   40,   65,  ...
0,  4,  18,  48,  101,  185,  ...
0,  6,  32,  99,  236,  481,  ...

Columns k=0..8 give A000007, A001935, A001936, A001937, A093160, A001939, A001940, A001941, A092877.
Main diagonal gives A296044.
Antidiagonal sums give A302020.
Cf. A296067.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i))/(1 - x^(2 i - 1)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[((1 - x^(4 i))/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[2^(-k/2) (EllipticTheta[2, 0, x]/(x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.
G.f. of column k: Product_{j>=1} ((1 - x^(4*j))/(1 - x^j))^k.
G.f. of column k: 2^(-k/2)*(theta_2(0,x)/(x^(1/8)*theta_2(Pi/4,sqrt(x))))^k, where theta_() is the Jacobi theta function.

A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

Original entry on oeis.org

1, 2, 5, 14, 38, 101, 275, 746, 2026, 5507, 14969, 40689, 110604, 300652, 817255, 2221528, 6038739, 16414993, 44620576, 121291299, 329703934, 896228212, 2436200862, 6622280533, 18001224835, 48932402358, 133012060152, 361564266077, 982833574297, 2671618645410

Offset: 1

Michel Lagneau, Jan 30 2010

nonn

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt and the complete elliptic integral CK is defined by CK(k) = F(1,sqrt(1-k^2)). We calculate the values of CK(k) with k = 1/p, p = 1,2,3, ... and we propose a very interesting property: a(n+1)/a(n) tends toward e = 2.7182818... when n tends to infinity. For example, a(8) / a(7) = 2.718281581; a(9) / a(8) = 2.7182817562.

			a(3) = 38 because floor(CK(1/37)) = 4 and floor(CK(1/38)) = 5.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
M. Abramowitz and I. Stegun, "Elliptic Integrals", Chapter 17 of Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).
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.

A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 56.
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]

Cf. elliptic functions: A001936, A002318, A001937, A001934, A001938, A002754, A001939, A001940, A001941, A002753, A006089, A004005.

a0:=1:for p from 1 to 1000 do:a:= evalf(EllipticCK(1/p)):if floor(a)=a0+1 then print(p):a0:=floor(a):else fi:od:

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt. CK is defined by CK(k) = F(1,sqrt(1-k^2)). a(n) is the n-th integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

cp's OEIS Frontend

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^n.

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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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A001941 Absolute values of coefficients of an elliptic function.

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A296068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j))/(1 - x^(2*j-1)))^k.

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Mathematica

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A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

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Maple

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A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

A296068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j))/(1 - x^(2j-1)))^k.