A382347 -id:A382347 - cp's OEIS frontend

A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3n+k)x).

1, 7, 146, 4578, 189144, 9660840, 586813968, 41283943344, 3299858098560, 295294500123840, 29242449106502400, 3174506423754019200, 374845813851886709760, 47828682507084551654400, 6557612642418946942310400, 961431335221085133398784000, 150095351600371197275428454400

Offset: 0

Seiichi Manyama, May 18 2025

nonn

Column k=3 of A382347.

Cf. A165675.

Table[SeriesCoefficient[Product[(1 + (3*n+k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 23 2025 *)

a(n) = sum(k=0, n, (k+1)*(3*n)^k*abs(stirling(n+1, k+1, 1)));

a(n) = A165675(4*n,3*n).
a(n) = Sum_{k=0..n} (k+1) * (3*n)^k * |Stirling1(n+1,k+1)|.
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-3*n,k)/(n+1-k).
a(n) = (4*n)!/(3*n)! * (1 + 3*n * Sum_{k=1..n} 1/(3*n+k)).
a(n) ~ log(4/3) * 2^(8*n+1) * n^(n+1) / (exp(n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, May 23 2025

A383678 a(n) = [x^n] Product_{k=0..n} (1 + (2n+k)x).

Original entry on oeis.org

1, 5, 74, 1650, 48504, 1763100, 76223664, 3817038960, 217177416576, 13834411290720, 975244141065600, 75366122480858880, 6335159176892851200, 575442172080117538560, 56165570794932257433600, 5862137958472255891200000, 651508569509254106827161600, 76814449419352043102473728000

Offset: 0

Seiichi Manyama, May 18 2025

nonn

Column k=2 of A382347.

Cf. A000142, A165675, A384024.

Table[SeriesCoefficient[Product[(1 + (2*n+k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 23 2025 *)

a(n) = sum(k=0, n, (k+1)*(2*n)^k*abs(stirling(n+1, k+1, 1)));

a(n) = A165675(3*n,2*n).
a(n) = Sum_{k=0..n} (k+1) * (2*n)^k * |Stirling1(n+1,k+1)|.
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-2*n,k)/(n+1-k).
a(n) = (3*n)!/(2*n)! * (1 + 2*n * Sum_{k=1..n} 1/(2*n+k)).
a(n) ~ log(3/2) * 3^(3*n + 1/2) * n^(n+1) / (exp(n) * 2^(2*n - 1/2)). - Vaclav Kotesovec, May 23 2025

A380707 a(n) = [x^n] Product_{k=0..n} (1 + (n^2+k)*x).

Original entry on oeis.org

1, 3, 74, 4578, 520024, 93638820, 24469489008, 8744195444880, 4093736159733120, 2430707964048640800, 1784480276787736636800, 1586934417435493101528960, 1680937045347184025188838400, 2091005717306225140393765228800, 3018259634660179964662904164915200

Offset: 0

Seiichi Manyama, May 18 2025

nonn

Main diagonal of A382347.

Cf. A165675.

Table[SeriesCoefficient[Product[1 + (n^2+k)*x, {k, 0, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 23 2025 *)

a(n) = sum(k=0, n, (k+1)*n^(2*k)*abs(stirling(n+1, k+1, 1)));

a(n) = A165675((n+1)*n,n^2).
a(n) = Sum_{k=0..n} (k+1) * n^(2*k) * |Stirling1(n+1,k+1)|.
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k * binomial(-n^2,k)/(n+1-k).
a(n) = ((n+1)*n)!/(n^2)! * (1 + n^2 * Sum_{k=1..n} 1/(n^2+k)).
a(n) ~ exp(1/2) * n^(2*n+1). - Vaclav Kotesovec, May 23 2025

cp's OEIS Frontend

A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3n+k)x).

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Formula

A383678 a(n) = [x^n] Product_{k=0..n} (1 + (2n+k)x).

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Crossrefs

Programs

Mathematica

PARI

Formula

A380707 a(n) = [x^n] Product_{k=0..n} (1 + (n^2+k)*x).

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Formula

cp's OEIS Frontend

A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3*n+k)*x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A383678 a(n) = [x^n] Product_{k=0..n} (1 + (2*n+k)*x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A380707 a(n) = [x^n] Product_{k=0..n} (1 + (n^2+k)*x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A382349 a(n) = [x^n] Product_{k=0..n} (1 + (3n+k)x).

A383678 a(n) = [x^n] Product_{k=0..n} (1 + (2n+k)x).