A010816 -id:A010816 - cp's OEIS frontend

A369983 Maximum of the absolute value of the coefficients of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

1, 3, 8, 15, 44, 50, 117, 186, 356, 561, 972, 1761, 3508, 5789, 10470, 19023, 35580, 62388, 113418, 205653, 376496, 674085, 1226181, 2211462, 4056220, 7287672, 13261764, 24005627, 43800562, 79033269, 143513301, 260061408, 473603594, 855436899, 1553736558, 2813222766

Offset: 0

Ilya Gutkovskiy, Feb 07 2024

nonn

Cf. A010816, A133871, A160089, A369709, A369711, A369790.

Table[Max[Abs[CoefficientList[Product[(1 - x^k)^3, {k, 1, n}], x]]], {n, 0, 35}]

a(n) = vecmax(apply(abs, Vec(prod(i=1, n, (1-x^i)^3)))); \\ Michel Marcus, Feb 07 2024

from collections import Counter
def A369983(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            d[j+k] -= 3*a
            d[j+2*k] += 3*a
            d[j+3*k] -= a
        c = d
    return max(map(abs,c.values())) # Chai Wah Wu, Feb 07 2024

A178737 Coefficients in expansion of Jacobi theta_1'''(0).

Original entry on oeis.org

1, -27, 0, 125, 0, 0, -343, 0, 0, 0, 729, 0, 0, 0, 0, -1331, 0, 0, 0, 0, 0, 2197, 0, 0, 0, 0, 0, 0, -3375, 0, 0, 0, 0, 0, 0, 0, 4913, 0, 0, 0, 0, 0, 0, 0, 0, -6859, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9261, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12167, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15625, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jun 08 2010

sign

			G.f. = 1 - 27*x + 125*x^3 - 343*x^6 + 729*x^10 - 1331*x^15 + 2197*x^21 + ...
G.f. = q - 27*q^9 + 125*q^25 - 343*q^49 + 729*q^81 - 1331*q^121 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A006352, A010816.

a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[x], (-1)^Quotient[x, 2] x^3, 0]]; (* Michael Somos, Mar 19 2017 *)

{a(n) = my(x); if( n<0, 0, if(issquare(8*n + 1, &x), (-1)^(x\2) * x^3))};

Given g.f. A(x), then q * A(q^8) = -1/2 * theta_1'''(0, q^4) where the Jacobi nome q = exp(-Pi * K' / K).
a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(3 * e/2) if p == 1 (mod 4), b(p^e) = (-p)^(3 * e/2) if p == 3 (mod 4).
Convolution of A006352 and A010816.
G.f.: Sum_{k>=0} (-1)^k * (2*k + 1)^3 * x^(k * (k+1) / 2).

A371552 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^3.

Original entry on oeis.org

1, -3, 3, 18, -57, -138, 246, 4281, 13383, -156906, -450822, -957729, 23375886, 289894875, -179027895, -3403581357, -174968380137, -419588974650, 4439383168602, 50400469832883, 1027067921064738, 428364930324489, -18456487538087145, -1019962180000311267

Offset: 0

Ilya Gutkovskiy, Mar 27 2024

sign

Cf. A010816, A032314, A185895, A371551.

nmax = 23; CoefficientList[Series[Product[(1 - x^k/k!)^3, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A380517 Absolute value of the minimum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

Original entry on oeis.org

1, 3, 6, 15, 24, 50, 81, 186, 305, 561, 972, 1761, 3129, 5789, 10470, 19023, 33549, 62388, 113418, 205653, 366198, 674085, 1226181, 2211462, 3953679, 7287672, 13261764, 24005627, 42998125, 79033269, 143513301, 260061408, 465444889, 855436899, 1553736558, 2813222766, 5052061560, 9250734231

Offset: 0

Ilya Gutkovskiy, Jan 26 2025

nonn

Cf. A010816, A086394, A369709, A369711, A369983, A380499.

p:= proc(n) option remember;
     `if`(n=0, 1, expand(p(n-1)*(1-x^n)^3))
    end:
a:= n-> abs(min(coeffs(p(n)))):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 26 2025

Table[Min[CoefficientList[Product[(1 - x^k)^3, {k, 1, n}], x]], {n, 0, 37}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, (1-x^k)^3)))); \\ Michel Marcus, Jan 26 2025

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A369983 Maximum of the absolute value of the coefficients of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

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A178737 Coefficients in expansion of Jacobi theta_1'''(0).

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A371552 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^3.

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A380517 Absolute value of the minimum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

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