A376945 -id:A376945 - cp's OEIS frontend

A376943 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)) Product_{j=1..k} (1 + x^j).

1, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 0, 8, 8, 8, 16, 8, 8, 8, 0, 16, 16, 16, 32, 32, 32, 32, 32, 16, 16, 48, 32, 32, 64, 64, 96, 96, 96, 96, 96, 96, 64, 128, 96, 96, 160, 128, 192, 256, 256, 256, 320, 320, 320, 320, 256, 384, 384, 320, 384, 384, 448, 576, 704, 640, 768, 896

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) * Product_{j=1..k} (1 + x^j), then a(n) ~ r^c * (1+r) * exp(sqrt((2*log(d)^2 + 8*b*log(d)*log(r) + 4*b*(2*b+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*b*(1+r))*n)), where r is the smallest positive real root of the equation d*r^(2*b)*(1+r) = 1.

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

Cf. A273065, A333179, A376944, A376945, A376947.

nmax = 80; CoefficientList[Series[Sum[2^k * x^(k*(k+1)) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 80; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^(2*k), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ r * (1+r) * exp(sqrt((2*log(2)^2 + 8*log(2)*log(r) + 12*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((3*r + 2)*n)), where r = ((46 - 6*sqrt(57))^(1/3) + (46 + 6*sqrt(57))^(1/3) - 2)/6 is the real root of the equation 2*r^2*(1+r) = 1 (A273065).

A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 12, 8, 8, 16, 24, 24, 24, 32, 32, 64, 64, 64, 80, 80, 112, 160, 160, 160, 224, 224, 256, 320, 416, 416, 480, 576, 576, 704, 768, 896, 1152, 1216, 1280, 1536, 1600, 1856, 2112, 2304, 2560, 3200, 3456, 3584, 4224, 4480, 5120, 5760, 6144, 6656, 7808, 9088

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

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

Cf. A032302, A053261, A376943, A376945.

nmax = 60; CoefficientList[Series[Sum[2^k * x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^k, {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ sqrt(1 + sqrt(3)) * exp(sqrt((2*log(2)^2 + 2*log(1 - sqrt(3)/2) * log(sqrt(3) - 1) + 4*polylog(2, sqrt(3) - 1) - Pi^2/3)*n)) / (4*3^(1/4)*sqrt(n)).

A376948 G.f.: Sum_{k>=0} 2^k * x^(k^2) / Product_{j=1..k} (1 - x^j).

Original entry on oeis.org

1, 2, 2, 2, 6, 6, 10, 10, 14, 22, 26, 34, 46, 54, 66, 82, 110, 126, 162, 194, 246, 286, 354, 410, 502, 606, 714, 842, 1014, 1190, 1418, 1658, 1950, 2278, 2666, 3090, 3646, 4198, 4882, 5634, 6558, 7534, 8754, 10002, 11558, 13230, 15218, 17322, 19910, 22702, 25914, 29466, 33606

Offset: 0

Vaclav Kotesovec, Oct 10 2024

nonn

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

Cf. A003114, A032302, A376945, A376947.

nmax = 80; CoefficientList[Series[Sum[2^k*x^(k^2) / Product[1-x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ (Pi^2/6 + log(2)^2)^(1/4) * exp(sqrt((Pi^2/3 + 2*log(2)^2)*n)) / (2^(3/4) * sqrt(3*Pi) * n^(3/4)).

cp's OEIS Frontend

A376943 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)) Product_{j=1..k} (1 + x^j).

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A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).

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A376948 G.f.: Sum_{k>=0} 2^k * x^(k^2) / Product_{j=1..k} (1 - x^j).

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cp's OEIS Frontend

A376943 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j).

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A376944 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j).

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A376948 G.f.: Sum_{k>=0} 2^k * x^(k^2) / Product_{j=1..k} (1 - x^j).

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A376943 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)) Product_{j=1..k} (1 + x^j).

A376944 G.f.: Sum_{k>=0} 2^k * x^(k(k+1)/2) Product_{j=1..k} (1 + x^j).