A007249 -id:A007249 - cp's OEIS frontend

A161196 Triangle read by rows generated from A007249, the convolution square root of A007191.

1, -12, -12, 66, 144, 66, -232, -792, -792, -232, 639, 2784, 4356, 2784, 639, -1596, -7668, -15312, -15312, -7668, -1596, 3774, 19152, 42174, 53824, 42174, 19152, 3774, -8328, -45288, -105336, -146248, -146248, -105336, -45288, -8328, 17283

Offset: 0

Gary W. Adamson & Alexander R. Povolotsky, Jun 06 2009

Row sums = A007191: (1, -24, 276, -2048, 11202,...)

			First few rows of the triangle =
1;
-12, -12;
66, 144, 66;
-232, -792, -792, -232;
639, 2784, 4356, 2784, 639;
-1596, -7668, -15312, -15312, -7668, -1596;
3774, 19152, 42174, 53824, 42174, 19152, 3774;
-8328, -45288, -105336, -148248, -148248, -105336, -45288, -8328;
17283, 99936, 249084, 370272, 408321, 370272, 249084, 99936, 17283;
-34520, -207396, -549648, -875568, -1019844, -1019844, -875568, -549648, -207396, -34520;
...

Cf. A007191, A007249

Triangle read by rows, self-convolution of A007249. Begin with M = an infinite lower triangular Toeplitz matrix with A007249 as every column. Let Q = a matrix with A007249: (1, -12, 66, -232,..) as the right border and the rest zeros. Triangle A161196 = M * Q.

A022577 Expansion of Product_{m>=1} (1+x^m)^12.

Original entry on oeis.org

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, 40944396120, 73680871326, 131211105208, 231355524048, 404110659732

Offset: 0

N. J. A. Sloane

nonn

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

			G.f. = 1 + 12*x + 78*x^2 + 376*x^3 + 1509*x^4 + 5316*x^5 + 16966*x^6 + ...
G.f. = q + 12*q^3 + 78*q^5 + 376*q^7 + 1509*q^9 + 5316*q^11 + 16966*q^13 + ...

Seiichi Manyama, 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

Column k=12 of A286335.

Cf. A000122, A000700, A007096, A007249, A010054, A121373, A124863.

Coefficients(&*[(1+x^m)^12:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

N:= 50:
G:= mul(1+x^m,m=1..N+1)^12:
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Feb 26 2018

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q)^(1/2) / (1 - m), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q) /(1-m)^(1/2), {q, 0, 2 n}]]; (* Michael Somos, Jul 22 2011 *)
CoefficientList[QPochhammer[-1, q]^12/4096+O[q]^30, q] (* Jean-François Alcover, Nov 27 2015 *)
With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^12, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 25 2018 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^k, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Jul 16 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^12, n))}; /* Michael Somos, Jul 16 2005 */

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^12)) \\ G. C. Greubel, Feb 25 2018

G.f.: Product_{k>=1} ( 1 + x^k )^12.
Expansion of chi(-x)^-12 in powers of x where chi() is a Ramanujan theta function.
Expansion of k^2 / (16 * q * k') in powers of q^2. - Michael Somos, Jul 22 2011
Expansion of q^(-1/2) * (k/4) / (1 - k^2) in powers of q. - Michael Somos, Jul 16 2005
Expansion of q^(-1/2) * (eta(q^2) / eta(q))^12 in powers of q. - Michael Somos, Jul 16 2005
Euler transform of period 2 sequence [12, 0, ...]. - Michael Somos, Jul 16 2005
Given g.f. A(x), then B(q) = (q * A(q^2))^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (4096*u*v + 48*u + 1)*v - u^2 . - Michael Somos, Jul 16 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007249. - Michael Somos, Jul 22 2011
A124863(n) = (-1)^n * a(n). A007096(4*n + 2) = 8 * a(n). Convolution inverse of A007249.
a(n) ~ exp(2 * Pi * sqrt(n)) / (128 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

More terms added by G. C. Greubel, Feb 25 2018

A007247 McKay-Thompson series of class 4B for the Monster group.

Original entry on oeis.org

1, 52, 834, 4760, 24703, 94980, 343998, 1077496, 3222915, 8844712, 23381058, 58359168, 141244796, 327974700, 742169724, 1627202744, 3490345477, 7301071680, 14987511560, 30138820888, 59623576440, 115928963656

Offset: 0

N. J. A. Sloane

nonn

			T4B = 1/q + 52*q + 834*q^3 + 4760*q^5 + 24703*q^7 + 94980*q^9 + ...

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

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.
Claude Duhr and Sara Maggio, Feynman integrals, elliptic integrals and two-parameter K3 surfaces, arXiv:2502.15326 [hep-th], 2025. See p. 12.
David Ford, John McKay, and Simon Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
John McKay and Hubertus 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

Cf. A007249, A022577.

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 *)

{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 */

{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 */

{ my(q='q+O('q^66), t=(eta(q)/eta(q^2))^12); Vec( t + 64*q/t ) } \\ Joerg Arndt, Apr 02 2017

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

A112142 McKay-Thompson series of class 8B for the Monster group.

Original entry on oeis.org

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, 201825420, 303257512

Offset: 0

Michael Somos, Aug 28 2005

nonn

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

			1 + 12*x + 66*x^2 + 232*x^3 + 639*x^4 + 1596*x^5 + 3774*x^6 + 8328*x^7 + ...
T8B = 1/q + 12*q + 66*q^3 + 232*q^5 + 639*q^7 + 1596*q^9 + 3774*q^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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

Cf. A007249, A124863.

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[ Product[ 1 + x^k, {k, 1, n, 2}]^-12, {x, 0, n}] (* Michael Somos, Jul 22 2011 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^12, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q^2]^2/(QP[q]*QP[q^4]))^12 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^12, n))}

Expansion of chi(q)^12 in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(1/2) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^12 in powers of q.
G.f.: Product_{k>0} (1 + (-x)^k)^-12 = Product_{k>0} (1 + x^(2*k - 1))^-12.
a(n) = (-1)^n * A007249(n). Convolution inverse of A124863.
G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(2*k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017

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

A052241 McKay-Thompson series of class 8C for Monster.

Original entry on oeis.org

1, 26, 79, 326, 755, 2106, 4460, 10284, 20165, 41640, 77352, 147902, 263019, 475516, 816065, 1413142, 2353446, 3936754, 6391091, 10390150, 16497734, 26184098, 40775677, 63394792, 97037170, 148178934, 223351867, 335704742, 499050461, 739575640, 1085723797

Offset: 0

N. J. A. Sloane, Nov 27 2000

nonn

Let q = exp(-Pi*sqrt(58)/4). Then 396 = B(q) = 1/q + 26*q^3 + ... + a(n)*q^(4*n-1) + ... - Michael Somos, Sep 30 2019

			G.f. = 1 + 26*x + 79*x^2 + 326*x^3 + 755*x^4 + 2106*x^5 + 4460*x^6 + ...
T8C = 1/q + 26*q^3 + 79*q^7 + 326*q^11 + 755*q^15 + 2106*q^19 + 4460*q^23 + ...

G. C. Greubel, 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, Emails to N. J. A. Sloane, 1993
Index entries for McKay-Thompson series for Monster simple group

Cf. A007247, A007249, A007267.

QP = QPochhammer; A = O[q]^40; A = (QP[q + A]/QP[q^2 + A])^12; s = Sqrt[A + 64*(q/A)]; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)
eta[q_] := q^(1/24)*QPochhammer[q]; e4D := q^(1/2)*(eta[q]/eta[q^2])^12;
T4B := e4D + 64*q/e4D; a[n_]:= SeriesCoefficient[Sqrt[(T4B /. {q -> q^2}) + O[q]^100], {q, 0, n}]; Table[a[n], {n, 0, 50}][[1 ;; ;; 2]] (* G. C. Greubel,Feb 13 2018 *)
a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, A}, A = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (A + 64/A)^(1/2), {q, 0, n - 1/4}]]; (* Michael Somos, Sep 30 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( sqrt(A + 64 * x / A), n))}; /* Michael Somos, Sep 01 2014 */

Expansion of 2 * q^(1/4) * ((k'^4 + 4*k^2) / (k'^2 * k))^(1/2) in powers of q. - Michael Somos, Sep 01 2014
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - (u*v - 12) * (u*v - 32)^2. - Michael Somos, Sep 01 2014
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, Sep 01 2014
Convolution square is A007247. Convolution fourth power is A007267.
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4)*n^(3/4)). - Vaclav Kotesovec, May 01 2017

A007250 McKay-Thompson series of class 4a for the Monster group.

Original entry on oeis.org

1, -76, -702, -5224, -23425, -98172, -336450, -1094152, -3188349, -8913752, -23247294, -58610304, -140786308, -328793172, -740736900, -1629664840, -3486187003, -7307990208, -14976155896, -30157221352, -59594117256, -115975615160, -222119374922, -419704427016

Offset: 0

N. J. A. Sloane

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

			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 - ...

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

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

Cf. A007242, A007249, A007260, A022577.

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

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 *)

{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 */

N=66; q='q+O('q^N); t=(eta(q)/eta(q^2))^12; Vec(t - 64*q/t) \\ Joerg Arndt, Mar 11 2017

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

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A161196 Triangle read by rows generated from A007249, the convolution square root of A007191.

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A022577 Expansion of Product_{m>=1} (1+x^m)^12.

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A007247 McKay-Thompson series of class 4B for the Monster group.

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A112142 McKay-Thompson series of class 8B for the Monster group.

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A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

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A052241 McKay-Thompson series of class 8C for Monster.

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A007250 McKay-Thompson series of class 4a for the Monster group.

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