A022598 -id:A022598 - 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

A339718 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^3.

Original entry on oeis.org

1, -3, -3, 3, -3, 6, -3, -4, 3, 6, -3, -3, -3, 6, 6, 9, -3, -3, -3, -3, 6, 6, -3, 9, 3, 6, -4, -3, -3, -3, -3, -12, 6, 6, 6, 3, -3, 6, 6, 9, -3, -3, -3, -3, -3, 6, -3, -18, 3, -3, 6, -3, -3, 9, 6, 9, 6, 6, -3, -3, -3, 6, -3, 15, 6, -3, -3, -3, 6, -3, -3, -15, -3, 6, -3, -3, 6, -3, -3, -18

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022598, A316441, A339335, A339717, A339719, A339720, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339335(n/d) * a(d).

a(p^k) = A022598(k) for prime p.

A107635 McKay-Thompson series of class 32a for the Monster group.

Original entry on oeis.org

1, 3, 3, 4, 9, 12, 15, 21, 30, 43, 54, 69, 94, 123, 153, 193, 252, 318, 391, 486, 609, 754, 918, 1119, 1376, 1680, 2019, 2432, 2946, 3540, 4220, 5034, 6015, 7157, 8463, 9999, 11835, 13956, 16374, 19206, 22542, 26376, 30750, 35829, 41745, 48526, 56250

Offset: 0

Michael Somos, May 18 2005

nonn

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

			G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 15*x^6 + 21*x^7 + ...
T32a = 1/q + 3*q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 15*q^47 + ...

G. C. Greubel, 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. A022598.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^2 / (QPochhammer[ x] QPochhammer[ x^4]))^3, {x, 0, n}]; (* Michael Somos, Jun 29 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^3, {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)))^3, n))};

Expansion of q^(1/8) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^3 in powers of q.
Expansion of chi(x)^3 = phi(x) / psi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 - u) - 9*u*v.
Euler transform of period 4 sequence [3, -3, 3, 0, ...].
G.f.: Product_{k>0} (1 + (-x)^k)^-3.
a(n) = (-1)^n * A022598(n).
a(n) ~ exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(3*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

A382343 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of 3 kinds.

Original entry on oeis.org

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

Offset: 0

Peter Dolland, Mar 27 2025

			Triangle starts:
 0 : [1]
 1 : [0, 3]
 2 : [0, 3,  6]
 3 : [0, 3,  9,  10]
 4 : [0, 3, 15,  18,  15]
 5 : [0, 3, 18,  36,  30,  21]
 6 : [0, 3, 24,  55,  66,  45,  28]
 7 : [0, 3, 27,  81, 114, 105,  63,  36]
 8 : [0, 3, 33, 108, 189, 195, 153,  84,  45]
 9 : [0, 3, 36, 145, 276, 348, 298, 210, 108,  55]
10 : [0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66]
...

Main diagonal gives A000217(n+1).
Row sums give A000716.
Cf. A008284 (1-kind), A382342 (2-kind).
Cf. A022598, A382025.

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:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 27 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j*b[n-i*j, Min[n-i*j, i-1]]*(j+2)*(j+1)/2, {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 30 2025, after Alois P. Heinz *)

from sympy import binomial
from sympy.utilities.iterables import partitions
kinds = 3 - 1   # the number of part kinds - 1
def t_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

T(n,n) = binomial(n + 2, 2) = A000217(n + 1).
T(n,1) = 3 for n >= 1.
T(n,k) = A382025(n,k) - A382025(n,k-1) for 1 <= k <= n.
Sum_{k=0..n} (-1)^k * T(n,k) = A022598(n). - Alois P. Heinz, Mar 27 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A339718 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^3.

Views

Author

Keywords

Crossrefs

Formula

A107635 McKay-Thompson series of class 32a for the Monster group.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382343 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of 3 kinds.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Formula