A319668 -id:A319668 - cp's OEIS frontend

A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

1, -1, -3, -3, -4, 3, 2, 19, 21, 32, 40, 45, 16, 8, -18, -125, -164, -291, -358, -530, -588, -724, -592, -675, -358, -207, 570, 1201, 2208, 3333, 4944, 6490, 8277, 10492, 11800, 13260, 14328, 14722, 12942, 12075, 5640, 603, -10444, -21120, -39360, -55876, -83488

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

sign

Convolution inverse of A329156.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003056, A032198, A077285, A088305, A104575, A319668, A329156, A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..46);  # Alois P. Heinz, Jul 18 2025

nmax = 46; CoefficientList[Series[Product[(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Product[(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k / (1 - x^k)^2).
G.f.: exp(-Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} 1 / B(x^k), where B(x) = g.f. of A088305.
a(n) = Sum_{k=0..A003056(n)} (-1)^k * A385001(n,k). - Alois P. Heinz, Jul 18 2025

A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2n-1)) (1 + x^n + x^(2n-1)) (1 - x^n - x^(2*n)).

Original entry on oeis.org

2, 5, 0, -1, 0, -18, -15, -23, -36, -25, -52, -35, -42, -8, 13, 4, 96, 100, 208, 227, 388, 434, 499, 709, 670, 837, 883, 1057, 775, 1044, 819, 643, 535, -78, -345, -970, -1494, -3017, -3142, -5078, -6102, -7711, -9410, -11406, -13148, -15353, -17831, -18841, -22708, -22955, -26117

Offset: 0

Paul D. Hanna, Feb 22 2024

sign

Consider function R(p,q,r) = Product_{n>=1} (1 + p^(n-1)*(q^n + r)) * (1 + p^n*(q^(n-1) + r)) * (1 - p^n*(q^n + r)) which yields Ramanujan's theta function at r = 0: R(p,q,0) = f(p,q) = Sum_{n=-oo..+oo} p^(n*(n-1)/2) * q^(n*(n+1)/2). This sequence arises from R(p,q,r) when p = x, q = x, and r = 1: A(x) = R(x,x,1).

			G.f.: A(x) = 2 + 5*x - x^3 - 18*x^5 - 15*x^6 - 23*x^7 - 36*x^8 - 25*x^9 - 52*x^10 - 35*x^11 - 42*x^12 - 8*x^13 + 13*x^14 + 4*x^15 + 96*x^16 + 100*x^17 + ...
where A(x) equals the infinite product
A(x) = (2 + x)*(1 + 2*x)*(1 - x - x^2) * (1 + x + x^3)*(1 + x^2 + x^3)*(1 - x^2 - x^4) * (1 + x^2 + x^5)*(1 + x^3 + x^5)*(1 - x^3 - x^6) * (1 + x^3 + x^7)*(1 + x^4 + x^7)*(1 - x^4 - x^8) * (1 + x^4 + x^9)*(1 + x^5 + x^9)*(1 - x^5 - x^10) * (1 + x^5 + x^11)*(1 + x^6 + x^11)*(1 - x^6 - x^12) * (1 + x^6 + x^13)*(1 + x^7 + x^13)*(1 - x^7 - x^14) * ...

Paul D. Hanna, Table of n, a(n) for n = 0..6400

Cf. A000726, A319668.

{a(n) = my(A);
A = prod(m=1,n+1, (1 + x^(m-1) + x^(2*m-1)) * (1 + x^m + x^(2*m-1)) * (1 - x^m - x^(2*m)) +x*O(x^n));
polcoeff(A,n)}
for(n=0,50, print1(a(n),", "))

cp's OEIS Frontend

A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

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A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2n-1)) (1 + x^n + x^(2n-1)) (1 - x^n - x^(2*n)).

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cp's OEIS Frontend

A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

Views

Author

Keywords

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Maple

Mathematica

Formula

A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2*n-1)) * (1 + x^n + x^(2*n-1)) * (1 - x^n - x^(2*n)).

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PARI

A370541 Expansion of g.f. A(x) = Product_{n>=1} (1 + x^(n-1) + x^(2n-1)) (1 + x^n + x^(2n-1)) (1 - x^n - x^(2*n)).