A107635 -id:A107635 - cp's OEIS frontend

A341241 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^3.

1, 0, 3, 3, 6, 9, 13, 21, 27, 40, 54, 75, 97, 129, 171, 220, 282, 360, 460, 576, 720, 896, 1116, 1374, 1682, 2061, 2517, 3050, 3684, 4449, 5354, 6414, 7656, 9135, 10875, 12891, 15243, 18015, 21243, 24966, 29286, 34326, 40156, 46851, 54573, 63509, 73794, 85551, 99035, 114555

Offset: 3

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A022598, A047655, A107635, A327381, A338463, A341221, A341243, A341244, A341245, A341246, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..52);  # Alois P. Heinz, Feb 07 2021

nmax = 52; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^3.

a(n) ~ A107635(n). - Vaclav Kotesovec, Feb 20 2021

A112150 McKay-Thompson series of class 16a for the Monster group.

Original entry on oeis.org

1, 6, 15, 26, 51, 102, 172, 276, 453, 728, 1128, 1698, 2539, 3780, 5505, 7882, 11238, 15918, 22259, 30810, 42438, 58110, 78909, 106392, 142770, 190698, 253179, 334266, 439581, 575784, 750613, 974316, 1260336, 1624702, 2086530, 2670162, 3406695, 4333590

Offset: 0

Michael Somos, Aug 28 2005

nonn

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

			G.f. = 1 + 6*x + 15*x^2 + 26*x^3 + 51*x^4 + 102*x^5 + 172*x^6 + 276*x^7 + ...
T16a = 1/q + 6*q^3 + 15*q^7 + 26*q^11 + 51*q^15 + 102*q^19 + 172*x^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for McKay-Thompson series for Monster simple group
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A022601, A107635, A112142.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^6, {x, 0, n}]; (* Michael Somos, Jul 03 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^6, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

{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)))^6, n))}; /* Michael Somos, Jul 03 2014 */

Expansion of chi(x)^6 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 03 2014
Expansion of q^(1/4) * 2 * (k(q) * k'(q))^(-1/2) in powers of q where k() is the elliptic modulus. - Michael Somos, Jul 03 2014
Expansion of q^(1/4) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 in powers of q. - Michael Somos, Jul 03 2014
Euler transform of period 4 sequence [ 6, -6, 6, 0, ...]. - Michael Somos, Jul 03 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) = (v^3 - u) * (u^3 - v) - 9*u*v * (-7 + 2*u*v). -  Michael Somos, Jul 03 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) =  f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 03 2014
G.f.: Product_{k>0} (1 + (-x)^k)^-6 = Product_{k>0} (1 + x^(2*k - 1))^6. - Michael Somos, Jul 03 2014
Convolution square is A112142. Convolution square of A107635. - Michael Somos, Jul 03 2014
a(n) = (-1)^n * A022601(n). - Michael Somos, Jul 03 2014
a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 29 2014

nonn

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

			G.f. = 1 + 3*q + 3*q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 2*q^6 + 3*q^8 + 5*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A093709, A107635, A121373, A244526, A244543.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 3*A[2] + 3*A[3];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {3, 0, -1, 0, 1, 0, -3, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 3, 0, -1, 0, 1, 0, -3][d%8 + 1]))};

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * (A + subst(A, x, x^2)) / 2, n))};

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 3*A[1] + 3*A[2];

Expansion of f(-q^3, -q^5)^2 * phi(q) / psi(-q) = f(-q^3, -q^5)^2 * chi(q)^3 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [3, -3, 1, 0, 1, -3, 3, -2, ...].
Moebius transform is period 8 sequence [3, 0, -1, 0, 1, 0, -3, 0, ...].
Convolution product of A244526 and A107635. Convolution product of A000122 and A093709.
a(n) = (A004018(n) + A033715(n)) / 2 = A244543(2*n).
a(2*n) = a(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 4*A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 + 1/sqrt(2))/2 = 2.681517... . - Amiram Eldar, Jun 08 2025

A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -2, 1, 4, -2, 0, 1, -1, 4, -2, -2, 4, 0, 0, 1, 2, -1, -2, 4, 0, -2, 0, -2, 5, 4, -4, 0, 4, 0, 0, 1, -4, 2, 0, -1, 4, -2, 0, 4, 2, 0, -2, -2, 4, 0, 0, -2, 1, 5, -4, 4, 4, -4, 0, 0, -4, 4, -2, 0, 4, 0, 0, 1, 8, -4, -2, 2, 0, 0, 0, -1, 2, 4, -2, -2, 0, 0

Offset: 1

Michael Somos, Jun 30 2014

sign

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

			G.f. = q + q^2 - 2*q^3 + q^4 + 4*q^5 - 2*q^6 + q^8 - q^9 + 4*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A107635, A121373, A143259, A243747, A244560.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] + A[3];

a[ n_] := If[ n < 1, 0, Sum[ {1, 0, -3, 0, 3, 0, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 0, -3, 0, 3, 0, -1][d%8 + 1]))};

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * (A - subst(A, x, x^2)) / 2, n))};

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] + A[2];

Expansion of q * f(-q, -q^7)^2 * phi(q) / psi(-q) = q * f(-q, -q^7)^2 * chi(q)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [1, -3, 3, 0, 3, -3, 1, -2, ...].
Moebius transform is period 8 sequence [1, 0, -3, 0, 3, 0, -1, 0, ...].
Convolution product of A244560 and A107635. Convolution product of A000122 and A143259.
a(n) = (A004018(n) - A033715(n)) / 2 = A243747(2*n).
a(2*n) = a(n). a(8*n + 3) = -2 * A033761(n). a(8*n + 5) = 4 * A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 - 1/sqrt(2))/2 = 0.460075... . - Amiram Eldar, Jun 08 2025

A382521 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of three kinds. Containers may be left empty.

Original entry on oeis.org

1, 3, 0, 6, 3, 0, 10, 9, 3, 0, 15, 18, 15, 3, 0, 21, 30, 36, 18, 3, 0, 28, 45, 66, 55, 24, 3, 0, 36, 63, 105, 114, 81, 27, 3, 0, 45, 84, 153, 195, 189, 108, 33, 3, 0, 55, 108, 210, 298, 348, 276, 145, 36, 3, 0, 66, 135, 276, 423, 558, 552, 405, 180, 42, 3, 0, 78, 165, 351, 570, 819, 936, 858, 549, 225, 45, 3, 0

Offset: 0

Peter Dolland, Mar 30 2025

			Array starts:
 0 : [1, 3,  6,  10,   15,   21,   28,    36,    45,    55,    66]
 1 : [0, 3,  9,  18,   30,   45,   63,    84,   108,   135,   165]
 2 : [0, 3, 15,  36,   66,  105,  153,   210,   276,   351,   435]
 3 : [0, 3, 18,  55,  114,  195,  298,   423,   570,   739,   930]
 4 : [0, 3, 24,  81,  189,  348,  558,   819,  1131,  1494,  1908]
 5 : [0, 3, 27, 108,  276,  552,  936,  1428,  2028,  2736,  3552]
 6 : [0, 3, 33, 145,  405,  858, 1532,  2427,  3543,  4880,  6438]
 7 : [0, 3, 36, 180,  549, 1248, 2340,  3861,  5811,  8190, 10998]
 8 : [0, 3, 42, 225,  741, 1785, 3510,  6000,  9300, 13410, 18330]
 9 : [0, 3, 45, 271,  957, 2451, 5051,  8967, 14307, 21126, 29424]
10 : [0, 3, 51, 324, 1227, 3312, 7137, 13125, 21552, 32553, 46194]
...

Antidiagonal sums give A000716.
Alternating antidiagonal sums give A107635.
Without empty containers: A382025.
Cf. A382343, A000217, 2 kinds: A382345.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+2)*(j+1)/2, j=0..n/i))))
    end:
A:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Mar 31 2025

from sympy import binomial
from sympy.utilities.iterables import partitions
def a_row(n, length=11) :
    if n == 0 : return [ binomial( k + 2, 2) for k in range( length) ]
    t = list( [0] * length)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( 2 + p[k], 2)
        if s > 0 :
            t[s] += fact
    a = list( [0] * length)
    for i in range( 1, length):
        for j in range( i, 0, -1):
            a[i] += t[j] * binomial( i - j + 2, 2)
    return a
for n in range(11): print(a_row(n))

A(0,k) = binomial(k + 2, 2) = A000217(k + 1).
A(1,k) = 3 * binomial(k + 1, 2).
A(n,1) = 3.
A(n,k) = Sum_{i=0..k} binomial(k + 2 - i, 2) * A382343(n,i) for k <= n.
A(n,k) = A382343(n+k,k).

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A341241 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^3.

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A112150 McKay-Thompson series of class 16a for the Monster group.

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A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A382521 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n unlabeled objects are distributed into k containers of three kinds. Containers may be left empty.

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