A000706 -id:A000706 - cp's OEIS frontend

A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

9, 9, 9, 9, 9, 6, 5, 1, 2, 6, 4, 5, 4, 8, 2, 2, 3, 4, 2, 9, 5, 0, 9, 8, 9, 1, 6, 8, 5, 2, 1, 1, 9, 2, 4, 7, 6, 5, 7, 5, 0, 9, 7, 8, 9, 3, 2, 6, 3, 4, 5, 8, 4, 8, 4, 4, 7, 7, 3, 2, 6, 9, 1, 0, 0, 4, 7, 2, 0, 1, 5, 2, 5, 7, 6, 7, 4, 4, 8, 2, 0, 3, 2, 6, 8, 9, 6, 2, 4, 9, 7, 3, 0, 1, 1, 9, 7, 2, 8, 1, 0, 8, 9

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99999651264548223429509891685211924765750978932634584844773269100472...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A000706, A292822, A292826.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-4*Pi]], 10, 103] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-4*Pi)) = exp(Pi/6)*Gamma(1/4)/(2^(11/8)*Pi^(3/4)).
A259150 = A259148 * exp(Pi/8)/sqrt(2). - Vaclav Kotesovec, Jul 03 2017

A287933 Coefficients in expansion of 1/E_8.

Original entry on oeis.org

1, -480, 168480, -52199040, 15119446560, -4198347132480, 1132514464199040, -299116847254053120, 77742157641008378400, -19951615350261029163360, 5068304275307482667436480, -1276700988345016720650917760

Offset: 0

Seiichi Manyama, Jun 17 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A008410 (E_8).

Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), this sequence (k=8), A285836 (k=10), A287964 (k=14).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n, where c = (262144 * Pi^24) / (81 * Gamma(1/3)^36) = 1.0839091249080051624370140889296742679583925822413671... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A288816 Coefficients in expansion of 1/E_2.

Original entry on oeis.org

1, 24, 648, 17376, 466152, 12505104, 335466144, 8999325120, 241418862504, 6476381979576, 173737557697968, 4660740989265312, 125030574027131424, 3354111390776151504, 89978497733627940672, 2413792838444465745216, 64753202305891291798824

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..700

Cf. A006352 (E_2).

Cf. this sequence (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), A285836 (k=10), A287964 (k=14).

G.f.: 1/(1 - 24*sum(k>=1, k*x^k/(1 - x^k))).

a(n) ~ c / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.900080074462078245744608120875628441926356101483729... - Vaclav Kotesovec, Jul 02 2017

A289567 Coefficients in expansion of 1/E_6^(1/2).

Original entry on oeis.org

1, 252, 103572, 46355904, 21754545876, 10493652271032, 5153897870227008, 2563741466120209536, 1287429765611338091988, 651251466581383330576956, 331360676706818772917367912, 169399388595923901462013678656

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), this sequence (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
E_6^(k/12): A289570 (k=-18), A000706 (k=-12), this sequence (k=-6), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A000706 (1/E_6), A288851, A289293 (E_6^(1/2)).

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 2^(5/2) * Gamma(3/4)^8 / (3*Pi^(5/2)) = 0.5480868931611627439175185425300450785609564636925943866686455998197... - Vaclav Kotesovec, Jul 09 2017, updated Mar 03 2018

A289637 Coefficients in expansion of -q*E'_6/E_6 where E_6 is the Eisenstein Series (A013973).

Original entry on oeis.org

504, 287280, 153540576, 82226602080, 44031499226064, 23578504122108096, 12626092121367162816, 6761166974864088760512, 3620548496603402008959384, 1938773508354916749345180960, 1038197035676506069321210300320

Offset: 1

Seiichi Manyama, Jul 09 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..366

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), this sequence (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A000706, A006352 (E_2), A013973 (E_6), A145095, A288851.

nmax = 20; Rest[CoefficientList[Series[504*x*Sum[k*DivisorSigma[5, k]*x^(k-1), {k, 1, nmax}]/(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

a(n) = Sum_{d|n} d * A288851(d).
a(n) = A288840(n)/2 + 12*A000203(n).
a(n) = -Sum_{k=1..n-1} A013973(k)*a(n-k) - A013973(n)*n.
G.f.: 1/2 * E_8/E_6 - 1/2 * E_2.
a(n) ~ exp(2*Pi*n). - Vaclav Kotesovec, Jul 09 2017

A285836 Coefficients in expansion of 1/E_10.

Original entry on oeis.org

1, 264, 205128, 95104416, 54329698632, 28308006715824, 15339873507244704, 8172566140980183360, 4385988806258507934024, 2346434028637391065282536, 1257009611855633134427201328, 672999598306502464042506285792

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A013974 (E_10).

Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), this sequence (k=10), A287964 (k=14).

a(n) ~ c * exp(2*Pi*n), where c = 128 * Gamma(3/4)^24 / (27 * Pi^6) = 0.648273189440897942951926047466605067667211940159693598407336163991191821438... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A287964 Coefficients in expansion of 1/E_14.

Original entry on oeis.org

1, 24, 197208, 47715936, 42451725912, 18015200386704, 10924205579505504, 5511557851517150400, 3039496830486964153944, 1604976096786795234999096, 865212805864755380070382608, 461861254217266216545148291872

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A058550 (E_14).

Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), A285836 (k=10), this sequence (k=14).

terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[1/Ei[14] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) ~ c * exp(2*Pi*n), where c = 512 * Gamma(3/4)^32 / (81 * Pi^8) = 0.445315094156993820198784527343140685155693441915367780875399576353998457... - Vaclav Kotesovec, Jul 02 2017, updated Mar 07 2018

A289570 Coefficients in expansion of 1/E_6^(3/2).

Original entry on oeis.org

1, 756, 501228, 311671584, 187266950892, 110121960638088, 63808586297102304, 36578013578688141504, 20797655630223547290348, 11749541312124028845092052, 6603568491137827506152966712, 3695593478842608407829235523808

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..365

E_6^(k/12): this sequence (k=-18), A000706 (k=-12), A289567 (k=-6), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A288851.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-3/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-3*A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) * sqrt(n), where c = 2^(17/2) * Gamma(3/4)^24 / (27 * Pi^(13/2)) = 1.0344943380746471723299237298670710161068814236907171661035... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

A289540 Coefficients in expansion of 1/E_6^(1/12).

Original entry on oeis.org

1, 42, 12852, 4780104, 1974512526, 863778376440, 391960077239304, 182430901827757632, 86505196617272556900, 41607881477457256661154, 20239469012268054187498440, 9935363620927698868439915544, 4914082482014906612773260362232

Offset: 0

Seiichi Manyama, Jul 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

E_6^(k/12): A289570 (k=-18), A000706 (k=-12), A289567 (k=-6), this sequence (k=-1), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A013973, A288851, A299503.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 26 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(5/12) * Gamma(3/4)^(4/3) / (3^(1/6) * Pi^(1/3) * Gamma(1/12)) = 0.08654217651555778130817946575840803466... - Vaclav Kotesovec, Jul 26 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018

A294182 Coefficients in expansion of E_4/E_6.

Original entry on oeis.org

1, 744, 393768, 210962976, 112966533672, 60492691156464, 32393330061359904, 17346357971979746880, 9288829947058862457384, 4974090926254339741926216, 2663584184830281769743846768, 1426327104764356980195826984032

Offset: 0

Seiichi Manyama, Feb 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A000706, A004009 (E_4), A013973 (E_6), A288261.

E_k/E_{k+2}: A294181 (k=2), this sequence (k=4), A294183 (k=6).

terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]/E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Convolution inverse of A288261.

a(n) ~ 8 * Gamma(3/4)^8 * exp(2*Pi*n) / (3*Pi^2). - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A287933 Coefficients in expansion of 1/E_8.

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A288816 Coefficients in expansion of 1/E_2.

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A289567 Coefficients in expansion of 1/E_6^(1/2).

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A289637 Coefficients in expansion of -q*E'_6/E_6 where E_6 is the Eisenstein Series (A013973).

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A285836 Coefficients in expansion of 1/E_10.

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A287964 Coefficients in expansion of 1/E_14.

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A289570 Coefficients in expansion of 1/E_6^(3/2).

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A289540 Coefficients in expansion of 1/E_6^(1/12).

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A294182 Coefficients in expansion of E_4/E_6.

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