A001936 -id:A001936 - cp's OEIS frontend

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

1, 1, 2, 4, 6, 11, 19, 30, 47, 76, 118, 181, 277, 417, 624, 929, 1367, 2001, 2913, 4210, 6056, 8665, 12328, 17466, 24640, 34600, 48395, 67442, 93625, 129520, 178588, 245429, 336252, 459324, 625613, 849762, 1151150, 1555378, 2096332, 2818630, 3780903, 5060240, 6757633, 9005106, 11975265

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A001935, A001936, A001937, A026007, A035528, A093160, A113415, A156616, A284474, A292037, A295832.

nmax = 44; CoefficientList[Series[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[x^k (1 - (-1)^k x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.
G.f.: exp(Sum_{k>=1} x^k*(1 - (-1)^k*x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3*(7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (12 * (7*Zeta(3))^(1/3)) - Pi^4 / (3024*Zeta(3)) - 1/24) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(11/8) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 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.

A319455 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^2.

Original entry on oeis.org

1, 2, 7, 14, 35, 66, 140, 252, 485, 840, 1512, 2534, 4347, 7084, 11705, 18622, 29862, 46522, 72779, 111310, 170534, 256586, 386101, 572488, 848050, 1240974, 1812979, 2621486, 3782669, 5410360, 7720237, 10932740, 15443120, 21669546, 30327570, 42196022, 58555543, 80832850

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

nonn

Convolution inverse of A002171.
Self-convolution of A002513.
Convolution of A000041 and A029862.
Euler transform of period 2 sequence [2, 4, ...].

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000041, A001934, A001936, A002171, A002513, A029862.

a:=series(mul(1/((1-x^k)*(1-x^(2*k)))^2,k=1..55),x=0,38): seq(coeff(a,x,n),n=0..37); # Paolo P. Lava, Apr 02 2019

nmax = 37; CoefficientList[Series[Product[1/((1 - x^k)*(1 - x^(2*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^2])^2, {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[2 Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

seq(n)={Vec(exp(2*sum(k=1, n, (4*sigma(k) - sigma(2*k))*x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Sep 19 2018

G.f.: Product_{k>=1} (1 + x^k)^2/(1 - x^(2*k))^4.
G.f.: exp(2*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(13/4)*n^(7/4)). - Vaclav Kotesovec, Sep 14 2021

A210656 Expansion of psi(x^3) * phi(-x)^2 / phi(-x^2) in power of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, 36, 130, 412, 1176, 3105, 7712, 18192, 41098, 89476, 188592, 386322, 771528, 1506036, 2879688, 5403628, 9966408, 18092599, 32366288, 57117660, 99526362, 171378512, 291841464, 491812740, 820684904, 1356794820, 2223458146, 3613417008, 5825889936

Offset: 0

Michael Somos, Mar 27 2012

nonn

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

			1 + 8*x + 36*x^2 + 130*x^3 + 412*x^4 + 1176*x^5 + 3105*x^6 + 7712*x^7 + ...
q^3 + 8*q^7 + 36*q^11 + 130*q^15 + 412*q^19 + 1176*q^23 + 3105*q^27 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001936.

nmax = 30; CoefficientList[Series[Product[((1 - x^(2*k))^4 * (1 - x^(6*k))^2 / ((1 - x^k)^4 * (1 - x^(3*k)) * (1 - x^(4*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^4 * eta(x^3 + A) * eta(x^ 4 + A)) )^2, n))}

Expansion of q^(-3/4) * ( eta(q^2)^4 * eta(q^6)^2 / (eta(q)^4 * eta(q^3) * eta(q^ 4)) )^2 in powers of q.
Euler transform of period 12 sequence [ 8, 0, 10, 2, 8, -2, 8, 2, 10, 0, 8, 0, ...].
A001936(9*n + 2) - A001936(n) = 4 * a(3*n). A001936(9*n + 5) = 4 * a(3*n + 1). A001936(9*n + 8) = 4 * a(3*n + 2).
a(n) ~ exp(sqrt(3*n)*Pi) / (32*sqrt(2)*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

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.

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A295831 Expansion of Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.

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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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A319455 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^2.

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A210656 Expansion of psi(x^3) * phi(-x)^2 / phi(-x^2) in power of x where phi(), psi() are Ramanujan theta functions.

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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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A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

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.

A319455 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^2.