[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a022577 -id:a022577
Displaying 1-8 of 8 results found. page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A007096 Expansion of theta_3 / theta_4.
(Formerly M3332) 		+10
21
1, 4, 8, 16, 32, 56, 96, 160, 256, 404, 624, 944, 1408, 2072, 3008, 4320, 6144, 8648, 12072, 16720, 22976, 31360, 42528, 57312, 76800, 102364, 135728, 179104, 235264, 307672, 400704, 519808, 671744, 864960, 1109904, 1419456, 1809568, 2299832 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair, Jan 27 2005
a(n) is the number of pairs of overpartitions into odd parts where the sum of all parts is equal to n. - Jeremy Lovejoy, Aug 29 2020
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.
Bernard L.S. Lin, Arithmetic properties of overpartition pairs into odd parts, Electronic J. Combin. 19, 2012, Paper 17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic, Mar 22 2005
Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.
Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A028939.
Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - Michael Somos, Jul 07 2005
Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.
G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.
A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019
EXAMPLE
G.f. = 1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
PROG
(PARI) {a(n) = my(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))}; /* Michael Somos, Jul 07 2005*/
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^2, n))}; /* Michael Somos, Jan 01 2006 */
CROSSREFS
Self-convolution of A080054. - Vladeta Jovovic, Mar 22 2005
Cf. A014969, A001936, A001938, A079006, A127391, A127392.
Cf. A022577, A080054, A097243, A123655.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Simon Plouffe
STATUS
approved
A007249 McKay-Thompson series of class 4D for the Monster group.
(Formerly M4846) 		+10
8
1, -12, 66, -232, 639, -1596, 3774, -8328, 17283, -34520, 66882, -125568, 229244, -409236, 716412, -1231048, 2079237, -3459264, 5677832, -9200232, 14729592, -23325752, 36567222, -56778888, 87369483, -133315692 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The convolution square root of A007191, and also the left and right borders of the triangle A161196. - Gary W. Adamson, Jun 06 2009
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
FORMULA
G.f.: Product_{m>=1} (1 + x^m)^(-12).
Expansion of chi(-x)^12 in powers of x where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022577. - Michael Somos, Jul 22 2011
a(n) = (-1)^n * A112142(n). (class 8B). Convolution inverse of A022577. - Michael Somos, Jul 22 2011
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-12*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Expansion of q^(1/2)*(eta(q)/eta(q^2))^12 in powers of q. - G. C. Greubel, Feb 13 2018
EXAMPLE
1 - 12*x + 66*x^2 - 232*x^3 + 639*x^4 - 1596*x^5 + 3774*x^6 + ...
T4D = 1/q - 12*q + 66*q^3 - 232*q^5 + 639*q^7 - 1596*q^9 + 3774*q^11 - ...
MATHEMATICA
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) / (m / 16 / q)^(1/2), {q, 0, n}]] (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/2) / (m / 16 / q), {q, 0, 2 n}]] (* Michael Somos, Jul 22 2011 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q]/QP[q^2])^12 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015, adapted from PARI *)
eta[q_]:=q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(1/2)*(eta[q]/eta[q^2])^12, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 13 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^12, n))} /* Michael Somos, Jul 22 2011 */
CROSSREFS
Cf. A007191, A022577, A112142.
Column k=12 of A286352.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
STATUS
approved
A007247 McKay-Thompson series of class 4B for the Monster group.
(Formerly M5305) 		+10
5
1, 52, 834, 4760, 24703, 94980, 343998, 1077496, 3222915, 8844712, 23381058, 58359168, 141244796, 327974700, 742169724, 1627202744, 3490345477, 7301071680, 14987511560, 30138820888, 59623576440, 115928963656 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..500 from Vincenzo Librandi)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Index entries for McKay-Thompson series for Monster simple group
FORMULA
Expansion of 4 * q * (1 + k'^2)^2 / (k' * k^2) in powers of q^2 where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
Expansion of 4 * q^(1/2) * (k'^4 + 4*k^2) / (k'^2 * k) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011
a(n) = A007249(n) + 64 * A022577(n - 1). - Michael Somos, Jul 22 2011
a(n) ~ exp(2*Pi*sqrt(n)) / (2*n^(3/4)). - Vaclav Kotesovec, Apr 01 2017
EXAMPLE
T4B = 1/q + 52*q + 834*q^3 + 4760*q^5 + 24703*q^7 + 94980*q^9 + ...
MATHEMATICA
a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, e}, e = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (e + 64 / e), {q, 0, n - 1/2}]] (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ 4 (2 - m)^2 / (m (1 - m)^(1/2)), {q, 0, 2 n - 1}]] (* Michael Somos, Jul 22 2011 *)
QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = A + 64*(q/A) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from 2nd PARI script *)
nmax = 30; CoefficientList[Series[64*x*Product[(1 + x^k)^12, {k, 1, nmax}] + Product[1/(1 + x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 01 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = prod( k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n))^12; polcoeff( A + 64 * x / A, n))} /* Michael Somos, Jul 22 2011 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( A + 64 * x / A, n))} /* Michael Somos, Nov 11 2006 */
(PARI) { my(q='q+O('q^66), t=(eta(q)/eta(q^2))^12); Vec( t + 64*q/t ) } \\ Joerg Arndt, Apr 02 2017
CROSSREFS
Cf. A007249, A022577.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
STATUS
approved
A022600 Expansion of Product_{m>=1} (1+q^m)^(-5). +10
4
1, -5, 10, -15, 30, -56, 85, -130, 205, -315, 465, -665, 960, -1380, 1925, -2651, 3660, -5020, 6775, -9070, 12126, -16115, 21220, -27765, 36235, -47101, 60810, -78115, 100105, -127825, 162391, -205530, 259475, -326565 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
PROG
(PARI) x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017
CROSSREFS
Cf. Related to Expansion of Product_{m>=1} (1+q^m)^k: A022627 (k=-32), A022626 (k=-31), A022625 (k=-30), A022624 (k=-29), A022623 (k=-28), A022622 (k=-27), A022621 (k=-26), A022620 (k=-25), A007191 (k=-24), A022618 (k=-23), A022617 (k=-22), A022616 (k=-21), A022615 (k=-20), A022614 (k=-19), A022613 (k=-18), A022612 (k=-17), A022611 (k=-16), A022610 (k=-15), A022609 (k=-14), A022608 (k=-13), A007249 (k=-12), A022606 (k=-11), A022605 (k=-10), A022604 (k=-9), A007259 (k=-8), A022602 (k=-7), A022601 (k=-6), this sequence (k=-5), A022599 (k=-4), A022598 (k=-3), A022597 (k=-2), A081362 (k=-1), A000009 (k=1), A022567 (k=2), A022568 (k=3), A022569 (k=4), A022570 (k=5), A022571 (k=6), A022572 (k=7), A022573 (k=8), A022574 (k=9), A022575 (k=10), A022576 (k=11), A022577 (k=12), A022578 (k=13), A022579 (k=14), A022580 (k=15), A022581 (k=16), A022582 (k=17), A022583 (k=18), A022584 (k=19), A022585 (k=20), A022586 (k=21), A022587 (k=22), A022588 (k=23), A014103 (k=24), A022589 (k=25), A022590 (k=26), A022591 (k=27), A022592 (k=28), A022593 (k=29), A022594 (k=30), A022595 (k=31), A022596 (k=32), A025233 (k=48).
Column k=5 of A286352.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
STATUS
approved
A189925 Expansion of theta_4/theta_3 in powers of q. +10
4
1, -4, 8, -16, 32, -56, 96, -160, 256, -404, 624, -944, 1408, -2072, 3008, -4320, 6144, -8648, 12072, -16720, 22976, -31360, 42528, -57312, 76800, -102364, 135728, -179104, 235264, -307672, 400704, -519808, 671744, -864960, 1109904 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In Baker [1890] page 94 is equation (1): sqrt(cos theta) = [[...]] = 1 - 4q + 8q^2 -[[...]] where cos theta = k'. - Michael Somos, Dec 31 2023
REFERENCES
Arthur L. Baker, Elliptic Functions, John Wiley & Sons, NY, 1890.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Berger, Relations between cusp forms on congruence and noncongruence groups, Proc. of the Amer. Math. Soc., vol. 128, (2000), 2869-2874. see p. 2870 equation (4).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q)^4 * eta(q^4)^2 / eta(q^2)^6 in powers of q.
Expansion of Jacobian elliptic function sqrt(k') in powers of q.
Expansion of phi(-q) / phi(q) = chi(-q)^2 / chi(q)^2 = psi(-q)^2 / psi(q)^2 = phi(-q)^2 / phi(-q^2)^2 = phi(-q^2)^2 / phi(q)^2 = chi(-q)^4 / chi(-q^2)^2 = chi(-q^2)^2 / chi(q)^4 = f(-q)^2 / f(q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -4, 2, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 * (u^2 + 1) - 2*u.
Unique solution to f(x^2)^(-2) = (f(x) + 1/f(x)) / 2 and f(0) = 1, f'(0) nonzero.
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A079006.
G.f.: theta_4 / theta_3 = (Sum_{k} (-x)^k^2)/(Sum_{k} x^k^2) = (Product_{k>0} ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2 / (1 - x^(4*k - 2)))^2.
Convolution inverse of A007096. a(n) = (-1)^n * A007096(n). a(2*n) = A014969(n). a(2*n + 1) = -4 * A093160(n). a(4*n) = A097243(n). a(4*n + 2) = 8*A022577(n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n))/(2^(5/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
G.f.: exp(-4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019
EXAMPLE
G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 32*q^4 - 56*q^5 + 96*q^6 - 160*q^7 + 256*q^8 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^2 / (1+x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
With[{nmax = 50}, CoefficientList[Series[4 QPochhammer[-1, x^2]^2/QPochhammer[-1, x]^4, {x, 0, nmax}], x]] (* Jan Mangaldan, Jan 04 2017 *)
With[{nmax = 50}, CoefficientList[Series[EllipticTheta[4, 0, x]/EllipticTheta[3, 0, x], {x, 0, nmax}], x]] (* Jan Mangaldan, Jan 04 2017 *)
a[ n_] := SeriesCoefficient[(1 - InverseEllipticNomeQ[x])^(1/4), {x, 0, n}]; (* Michael Somos, Dec 31 2023 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3 )^2, n))};
CROSSREFS
Cf. A007096, A014969, A079006, A093160.
KEYWORD
sign
AUTHOR
Michael Somos, May 01 2011
STATUS
approved
A124863 Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function. +10
3
1, -12, 78, -376, 1509, -5316, 16966, -50088, 138738, -364284, 913824, -2203368, 5130999, -11585208, 25444278, -54504160, 114133296, -234091152, 471062830, -931388232, 1811754522, -3471186596, 6556994502, -12222818640, 22502406793 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * (k * k') / 4 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2)^2)^12 in powers of q.
Euler transform of period 4 sequence [ -12, 12, -12, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011
G.f.: Product_{k>0} (1 + (-x)^k)^12 = Product_{k>0} 1/(1 + x^(2*k - 1))^12. [corrected by Vaclav Kotesovec, Nov 16 2017]
a(n) = (-1)^n * A022577(n). Convolution inverse of A112142. Convolution square is A100130.
G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(-x)^(k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)) / (128*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
EXAMPLE
G.f. = 1 - 12*x + 78*x^2 - 376*x^3 + 1509*x^4 - 5316*x^5 + 16966*x^6 - 50088*x^7 + ...
G.f. = q - 12*q^3 + 78*q^5 - 376*q^7 + 1509*q^9 - 5316*q^11 + 16966*q^13 - 50088*q^15 + ...
MATHEMATICA
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m/16/q)^(-1/2), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[1/Product[1 + q^k, {k, 1, n, 2}]^12, {q, 0, n}]; (* Michael Somos, Jul 22 2011, fixed by Vaclav Kotesovec, Nov 16 2017 *)
nmax = 30; CoefficientList[Series[Product[1/(1 + x^(2*k - 1))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^12, n))};
CROSSREFS
Cf. A022577, A100130, A112142.
KEYWORD
sign
AUTHOR
Michael Somos, Nov 10 2006
STATUS
approved
A007250 McKay-Thompson series of class 4a for the Monster group.
(Formerly M5353) 		+10
2
1, -76, -702, -5224, -23425, -98172, -336450, -1094152, -3188349, -8913752, -23247294, -58610304, -140786308, -328793172, -740736900, -1629664840, -3486187003, -7307990208, -14976155896, -30157221352, -59594117256, -115975615160, -222119374922, -419704427016 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A more correct name would be: Expansion of replicable function of class 4a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..499 from G. A. Edgar)
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Index entries for McKay-Thompson series for Monster simple group
FORMULA
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = - f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011
a(n) = A007249(n) - 64 * A022577(n-1).
Expansion of q^(1/2) * ((eta(q) / eta(q^2))^12 - 64*(eta(q^2) / eta(q))^12) in powers of q. - G. A. Edgar, Mar 10 2017
EXAMPLE
G.f. = 1 - 76*x - 702*x^2 - 5224*x^3 - 23425*x^4 - 98172*x^5 - 336450*x^6 + ...
T4a = 1/q - 76*q - 702*q^3 - 5224*q^5 - 23425*q^7 - 98172*q^9 - ...
MAPLE
A022577L := proc(n)
mul((1+x^m)^12, m=1..n+1) ;
taylor(%, x=0, n+1) ;
gfun[seriestolist](%) ;
end proc:
A007249L := proc(n)
if n = 0 then
0 ;
else
mul(1/(1+x^m)^12, m=1..n+1) ;
taylor(%, x=0, n+1) ;
gfun[seriestolist](%) ;
end if;
end proc:
a022577 := A022577L(80) ;
a007249 := A007249L(80) ;
printf("1, ");
for i from 1 to 78 do
printf("%d, ", op(i+1, a007249)-64*op(i, a022577) );
end do: # R. J. Mathar, Sep 30 2011
MATHEMATICA
a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, e}, e = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (e - 64 / e) q^(1/2), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = A - 64*(q/A) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
nmax = 30; CoefficientList[Series[Product[((1-x^k) / (1-x^(2*k)))^12, {k, 1, nmax}] - 64*x*Product[((1-x^(2*k)) / (1-x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( A - 64 * x / A, n))}; /* Michael Somos, Jul 22 2011 */
(PARI) N=66; q='q+O('q^N); t=(eta(q)/eta(q^2))^12; Vec(t - 64*q/t) \\ Joerg Arndt, Mar 11 2017
CROSSREFS
Cf. A007242, A007249, A007260, A022577.
KEYWORD
sign,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A260145 Expansion of x * (psi(x^4) / phi(x))^2 in powers of x where phi(), psi() are Ramanujan theta functions. +10
1
1, -4, 12, -32, 78, -176, 376, -768, 1509, -2872, 5316, -9600, 16966, -29408, 50088, -83968, 138738, -226196, 364284, -580032, 913824, -1425552, 2203368, -3376128, 5130999, -7738136, 11585208, -17225472, 25444278, -37350816, 54504160, -79085568, 114133296 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
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
FORMULA
Expansion of (eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5)^2 in powers of q.
Euler transform of period 8 sequence [ -4, 6, -4, 4, -4, 6, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A210067.
G.f.: x * Product_{k>0} ( 1 + x^(2*k))^6 * (1 + x^(4*k))^4 / (1 + x^k)^4.
a(n) = -(-1)^n * A107035(n). -4 * a(n) = A210066(n) unless n=0. -8 * a(n) = A139820(n) unless n=0.
a(2*n) = -4 * A092877(n). a(2*n + 1) = A022577(n). a(4*n) = -32 * A014103(n).
Convolution square of A210063. Convolution inverse of A131125.
a(n) ~ -(-1)^n * exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
EXAMPLE
G.f. = x - 4*x^2 + 12*x^3 - 32*x^4 + 78*x^5 - 176*x^6 + 376*x^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^2]^2 / EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5)^2, n))};
CROSSREFS
Cf. A014103, A022577, A092877, A107035, A131125, A139820, A210063, A210066, A210067.
KEYWORD
sign
AUTHOR
Michael Somos, Jul 17 2015
STATUS
approved
page 1

Search completed in 0.015 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 29 21:34 EDT 2024. Contains 375518 sequences. (Running on oeis4.)