A376812 -id:A376812 - cp's OEIS frontend

A376815 Decimal expansion of a constant related to the asymptotics of A376812.

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

Offset: 1

Vaclav Kotesovec, Oct 05 2024

			4.18362308231501037592434207471436289895638697707035887857832710...

Cf. A333198, A376621, A376658, A376659, A376660, A376812.

Cf. A088559.

RealDigits[E^Sqrt[3*Log[r]^2 + 8*PolyLog[2, r^(1/2)] - 2*Pi^2/3] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Equals limit_{n->infinity} A376812(n)^(1/sqrt(n)).
Equals A376660^2. - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2 + 8*polylog(2, r^(1/2)) - 2*Pi^2/3)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A216222 Counting a set of restricted partitions.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 4, 3, 3, 3, 3, 6, 7, 8, 10, 9, 9, 9, 9, 11, 13, 16, 20, 22, 25, 28, 27, 28, 29, 30, 32, 35, 40, 45, 53, 60, 67, 73, 79, 85, 87, 92, 95, 98, 105, 111, 120, 132, 145, 160, 178, 196, 212, 231, 247, 263, 280, 291, 305, 319, 334, 352, 371, 393

Offset: 0

David S. Newman, Mar 13 2013

nonn

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

Cf. A001622, A306734, A376542, A376580, A376812, A376813.

Take[CoefficientList[Sum[x^(k^2)*Product[1 + x^i, {i, k}]^2, {k, 0, 7}], x], 63] (* Giovanni Resta, Mar 13 2013 *)
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^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] (* Vaclav Kotesovec, Oct 09 2024 *)

G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j)^2 = 1 +x^1*(1+x)^2 +x^4*(1+x)^2*(1+x^2)^2 +...+ x^k^2*(1+x)^2*(1+x^2)^2*(1+x^3)^2*...*(1+x^k)^2+...

a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (4*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 29 2024

a(14)-a(62) from Giovanni Resta, Mar 13 2013

A376813 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 2, 2, 3, 6, 7, 8, 10, 8, 8, 8, 6, 8, 10, 12, 16, 20, 22, 24, 27, 26, 25, 26, 25, 26, 29, 32, 37, 44, 52, 58, 66, 72, 76, 82, 82, 84, 87, 88, 91, 96, 103, 112, 126, 138, 154, 174, 190, 208, 225, 238, 253, 268, 275, 284, 296, 304

Offset: 0

Vaclav Kotesovec, Oct 05 2024

nonn

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

Cf. A333179, A376623, A376812, A216222.

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

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

a(n) ~ phi^(1/2) * exp(Pi*sqrt(2*n/15)) / (4 * 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Original entry on oeis.org

1, 1, 4, 9, 16, 28, 49, 84, 140, 228, 361, 560, 856, 1288, 1916, 2821, 4108, 5928, 8480, 12024, 16920, 23637, 32788, 45196, 61928, 84368, 114332, 154160, 206857, 276308, 367476, 486680, 641996, 843656, 1104592, 1441168, 1873965, 2428816, 3138132, 4042408, 5192132

Offset: 0

Vaclav Kotesovec, Oct 06 2024

nonn

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

Cf. A001935, A143184, A192918, A376812, A376852, A376854.

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

a(n) ~ c * exp(sqrt(8*n*(log(r)^2 + polylog(2,r) - polylog(2,-r)))), where r = A192918 = 0.54368901269207636157... is the real root of the equation r*(1+r^2) = (1-r^2) and c = 0.0643033662740307713580663125340126524175...

cp's OEIS Frontend

A376815 Decimal expansion of a constant related to the asymptotics of A376812.

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A216222 Counting a set of restricted partitions.

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A376813 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^j)^2.

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A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

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cp's OEIS Frontend

A376815 Decimal expansion of a constant related to the asymptotics of A376812.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A216222 Counting a set of restricted partitions.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A376813 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)^2.

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A376853 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

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Mathematica

Formula

A376813 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^j)^2.

A376853 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.