A306734 -id:A306734 - cp's OEIS frontend

A350738 Expansion of Sum_{k>=0} (-1)^k * x^(k^2) * Product_{j=1..k} (1+x^j).

1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -2, -1, -1, -1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 0, 0, -1, -2, -2, -3, -3, -3, -3, -3, -3, -1, -1, 0, 1, 1, 3, 4, 4, 4, 5, 5, 5, 5, 3, 3, 3, 1, 0, -1, -3, -4, -4, -6, -7, -7, -8, -8, -8, -7, -7, -6, -5, -4, -2, -1, 1, 3, 5, 6, 8, 9, 10, 12, 13, 13, 12, 13, 12, 11, 11, 9, 7, 5, 3, 0

Offset: 0

Seiichi Manyama, Jan 12 2022

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A039924, A306734, A350310, A350737.

my(N=99, x='x+O('x^N)); Vec(sum(k=0, sqrtint(N), (-1)^k*x^k^2*prod(j=1, k, 1+x^j)))

from math import prod, isqrt
from sympy import Poly
from sympy.abc import x
def A350738(n): return Poly(sum((-1 if k % 2 else 1)*x**(k**2)*prod(1+x**j for j in range(1,k+1)) for k in range(isqrt(n+1)+1))).all_coeffs()[-n-1] # Chai Wah Wu, Jan 14 2022

A377080 G.f.: Sum_{k>=1} x^(2k^2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4

Offset: 0

Vaclav Kotesovec, Oct 15 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A306734, A350890, A350893, A377081.

Cf. A230152.

nmax = 150; CoefficientList[Series[Sum[x^(2*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/2]}], {x, 0, nmax}], x]

a(n) ~ (1+r) * exp(sqrt((40*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((4 + 5*r)*n)), where r = A230152 = 0.856674883854502874852324... is the real root of the equation r^4*(1+r) = 1.

A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2

Offset: 0

Vaclav Kotesovec, Oct 15 2024

In general, if m > 0 and g.f. = Sum_{k>=1} x^(m*k^2) * Product_{j=1..k} (1 + x^j), then a(n) ~ (1+r) * exp(sqrt((4*m*(2*m+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*m*(1+r))*n)), where r is the smallest positive real root of the equation r^(2*m)*(1+r) = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A306734 (m=1), A377080 (m=2).

Cf. A230154, A350890, A350894.

nmax = 200; CoefficientList[Series[Sum[x^(3*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/3]}], {x, 0, nmax}], x]

a(n) ~ (1+r) * exp(sqrt((84*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((6 + 7*r)*n)), where r = A230154 = 0.898653712628699293260875722... is the real root of the equation r^6*(1+r) = 1.

A376632 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 3, 2, 3, 3, 2, 4, 3, 4, 2, 4, 2, 5, 3, 5, 2, 5, 3, 5, 4, 4, 5, 5, 5, 5, 6, 4, 7, 4, 7, 4, 8, 4, 8, 5, 8, 6, 8, 6, 9, 7, 8, 8, 8, 9, 8, 10

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A306734, A376627, A376631.

nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ sqrt(1 + 3/sqrt(5)) * exp(Pi*sqrt(n/30)) / (4*sqrt(n)).

A306911 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j)^j.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 2, 2, 1, 2, 1, 2, 5, 4, 7, 9, 7, 10, 9, 9, 13, 13, 18, 27, 31, 42, 53, 61, 71, 83, 95, 98, 115, 131, 147, 176, 207, 258, 313, 395, 481, 581, 721, 848, 1014, 1179, 1367, 1586, 1804, 2064, 2338, 2698, 3083, 3559, 4142, 4819, 5732, 6768, 8036, 9582, 11426

Offset: 0

Ilya Gutkovskiy, Mar 16 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..4000

Cf. A306708, A306734, A318771.

N:= 100:
S:= series(add(x^(k^2)*mul((1+x^j)^j,j=1..min(k,N-k^2)),k=0..floor(sqrt(N))), x, N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Apr 10 2019

nmax = 60; CoefficientList[Series[Sum[x^(k^2) Product[(1 + x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306910 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + j*x^j).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 5, 3, 6, 6, 1, 1, 2, 5, 7, 10, 14, 20, 12, 25, 25, 2, 5, 7, 15, 19, 30, 37, 59, 74, 71, 101, 62, 125, 127, 15, 25, 36, 49, 89, 116, 160, 214, 241, 343, 476, 449, 427, 615, 385, 763, 776, 103, 151, 209, 319, 415, 594, 818, 1068, 1234, 1725

Offset: 0

Ilya Gutkovskiy, Mar 16 2019

nonn

Cf. A306707, A306734, A318770.

nmax = 67; CoefficientList[Series[Sum[x^(k^2) Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A350738 Expansion of Sum_{k>=0} (-1)^k * x^(k^2) * Product_{j=1..k} (1+x^j).

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A377080 G.f.: Sum_{k>=1} x^(2*k^2) * Product_{j=1..k} (1 + x^j).

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A377081 G.f.: Sum_{k>=1} x^(3*k^2) * Product_{j=1..k} (1 + x^j).

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A376632 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

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A306911 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j)^j.

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Maple

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A306910 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + j*x^j).

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Mathematica

A377080 G.f.: Sum_{k>=1} x^(2k^2) Product_{j=1..k} (1 + x^j).

A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).