A022599 -id:A022599 - cp's OEIS frontend

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

1, 0, 4, 4, 10, 16, 26, 44, 63, 100, 144, 212, 297, 420, 584, 796, 1081, 1452, 1940, 2556, 3355, 4372, 5668, 7288, 9327, 11892, 15076, 19012, 23884, 29904, 37276, 46284, 57276, 70680, 86918, 106528, 130220, 158784, 193054, 234076, 283178, 341824, 411616, 494512, 592933

Offset: 4

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A000700, A001482, A022599, A112160, A327382, A338463, A341222, A341241, A341244, A341245, A341246, A341247, A341251.

Column k=4 of A341279.

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, 4):
seq(a(n), n=4..48);  # Alois P. Heinz, Feb 07 2021

nmax = 48; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^4.

a(n) ~ A112160(n). - Vaclav Kotesovec, Feb 20 2021

A339719 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^4.

Original entry on oeis.org

1, -4, -4, 6, -4, 12, -4, -8, 6, 12, -4, -12, -4, 12, 12, 17, -4, -12, -4, -12, 12, 12, -4, 20, 6, 12, -8, -12, -4, -20, -4, -28, 12, 12, 12, 10, -4, 12, 12, 20, -4, -20, -4, -12, -12, 12, -4, -48, 6, -12, 12, -12, -4, 20, 12, 20, 12, 12, -4, 4, -4, 12, -12, 38, 12, -20, -4, -12, 12, -20

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022599, A316441, A339336, A339717, A339718, A339720, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339336(n/d) * a(d).

a(p^k) = A022599(k) for prime p.

A022569 Expansion of Product_{m>=1} (1+x^m)^4.

Original entry on oeis.org

1, 4, 10, 24, 51, 100, 190, 344, 601, 1024, 1702, 2768, 4422, 6948, 10752, 16424, 24782, 36972, 54602, 79872, 115805, 166540, 237664, 336720, 473856, 662596, 920934, 1272728, 1749407, 2392268, 3255410, 4409344, 5945730, 7983388, 10675712, 14220240, 18870672, 24951740, 32878114

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 + 4*x + 10*x^2 + 24*x^3 + 51*x^4 + 100*x^5 + 190*x^6 + 344*x^7 + ...
G.f. = q + 4*q^7 + 10*q^13 + 24*q^19 + 51*q^25 + 100*q^31 + 190*q^37 + 344*q^43 + ...

Seiichi Manyama, 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. 8.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A022599.

Column k=4 of A286335.

Coefficients(&*[(1+x^m)^4:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-4, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^4, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^4, n))}; /* Michael Somos, Apr 26 2008 */

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^4)) \\ G. C. Greubel, Feb 26 2018

Expansion of q^(-1/6) * (eta(q^2) / eta(q))^4 in powers of q.
Expansion of chi(-q)^(-4) in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 2 sequence [ 4, 0, ...]. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(q) = (A(q^6) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v * (1 + 16 * u * v) - u^2. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(x) = A(q^6) * q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) - 4 * w^2 * (u^2 + v). - Michael Somos, Apr 26 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022599.
G.f.: Product_{k>0} (1 + x^k)^4.
Convolution inverse of A022599.
G.f.: T(0)/x, where T(k) = 1 - 1/(1 - (1+(x)^(k+1))^4/((1+(x)^(k+1))^4 - 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2013
a(n) ~ exp(2 * Pi * sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)) * (1 + (Pi/(6*sqrt(3)) - 3*sqrt(3)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 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

A382344 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 4 kinds.

Original entry on oeis.org

1, 0, 4, 0, 4, 10, 0, 4, 16, 20, 0, 4, 26, 40, 35, 0, 4, 32, 80, 80, 56, 0, 4, 42, 124, 180, 140, 84, 0, 4, 48, 184, 320, 340, 224, 120, 0, 4, 58, 248, 535, 660, 574, 336, 165, 0, 4, 64, 332, 800, 1200, 1184, 896, 480, 220, 0, 4, 74, 416, 1176, 1956, 2284, 1932, 1320, 660, 286

Offset: 0

Peter Dolland, Mar 28 2025

			Triangle starts:
 0 : [1]
 1 : [0, 4]
 2 : [0, 4, 10]
 3 : [0, 4, 16,  20]
 4 : [0, 4, 26,  40,   35]
 5 : [0, 4, 32,  80,   80,   56]
 6 : [0, 4, 42, 124,  180,  140,   84]
 7 : [0, 4, 48, 184,  320,  340,  224,  120]
 8 : [0, 4, 58, 248,  535,  660,  574,  336,  165]
 9 : [0, 4, 64, 332,  800, 1200, 1184,  896,  480, 220]
10 : [0, 4, 74, 416, 1176, 1956, 2284, 1932, 1320, 660, 286]
...

Main diagonal gives A000292(n+1).
Row sums give A023003.
Cf. A008284 (1-kind), A382342 (2-kind), A382343 (3-kind).
Cf. A022599, A382041.

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))*binomial(j+3, 3), 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 28 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]]*Binomial[j+3, 3], {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, Aug 07 2025, after Alois P. Heinz *)

from sympy import binomial
from sympy.utilities.iterables import partitions
kinds = 4 - 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 + 3, 3) = A000292(n + 1).
T(n,1) = 4 for n >= 1.
T(n,k) = A382041(n,k) - A382041(n,k-1) for 1 <= k <= n.
Sum_{k=0..n} (-1)^k * T(n,k) = A022599(n). - Alois P. Heinz, Mar 28 2025

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A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

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A339719 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^4.

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A022569 Expansion of Product_{m>=1} (1+x^m)^4.

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

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A382344 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 4 kinds.

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