A344062 -id:A344062 - cp's OEIS frontend

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

1, -1, -3, -6, -18, -27, -108, -81, -486, 0, -1458, 8748, -6561, 118098, 118098, 1003833, 1417176, 11691702, 9565938, 105225318, 114791256, 746143164, 1076168025, 7231849128, 2324522934, 58113073350, 45328197213, 334731302496, 146444944842, 3263630199336, -3012581722464

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A003056, A032308, A242587, A261582, A292128, A300579, A337299, A344062, A352762.

nmax = 30; CoefficientList[Series[Product[(1 - 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 30}]

a(n) = Sum_{k=0..A003056(n)} (-1)^k * q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 10, 39, 390, 1521, 7830, 49518, 207360, 951102, 4264650, 22185657, 89579520, 401428224, 1676401110, 7172977275, 31972081050, 130330236546, 537393139200, 2213787635712, 8988968449530, 36073295687070, 150459195064320, 590262148332288, 2362876271009370, 9314694641056095

Offset: 0

Vaclav Kotesovec, Mar 02 2024

nonn

In general, if d >= 1 and g.f. = Product_{k>=1} (1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k), then a(n) ~ d^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(d)^2))) * (Pi^2/3 + log(d)^2)^(1/4) / (2^(5/4) * sqrt(Pi) * (d+1) * n^(3/4)).

Cf. A032308, A344062.

Cf. A022567 (d=1), A370761 (d=2).

nmax = 25; CoefficientList[Series[Product[(1+3^(k+1)*x^k)*(1+3^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(3)^2))) * (Pi^2/3 + log(3)^2)^(1/4) / (2^(13/4) * sqrt(Pi) * n^(3/4)).

cp's OEIS Frontend

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

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A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).

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cp's OEIS Frontend

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)*x^k) * (1 + 3^(k-1)*x^k).

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Formula

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).