A381512 -id:A381512 - cp's OEIS frontend

A348082 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1)^2 * x).

1, 10, 5082, 8187608, 27350858986, 155829826875450, 1352947132455198360, 16634466165612256277904, 275064994463136775255491210, 5887721317348514340055453080350, 158391364687146632772523433272637642, 5231238431447353406197858182627897590880

Offset: 0

Seiichi Manyama, Sep 27 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..165

Cf. A008958, A346543, A348081, A381512.

Table[SeriesCoefficient[Product[1/(1 - (2*k-1)^2*x), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)^2*x+x*O(x^n)), n);

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 4 * (2+r)^6 / (r^2 * (4+r)^2) = 314.10823271731893046905221731661671603309238326838259911942334135410817..., where r = 0.329482909104375658581668801636329590897344041849... is the root of the equation 4+r = r*exp(6/(2+r)) and c = 1/(2*Pi^(3/2)*sqrt(8/(r*(4 + r)) - 1)) = 0.041829340046147280338756273441751288807538817430199591424694081075... - Vaclav Kotesovec, Oct 16 2021, updated May 17 2025

a(n) = A381512(n,4*n-1) = (1/(2^(4*n-2)*(4*n-1)!)) * Sum_{k=0..2*n-1} (-1)^k * (4*n-1-2*k)^(6*n-1) * binomial(4*n-1,k) for n > 0. - Seiichi Manyama, May 16 2025

A383837 a(n) = (3n)!/n! [x^(3*n)] sinh(x)^n.

Original entry on oeis.org

1, 1, 16, 820, 87296, 15857205, 4390088704, 1721255653656, 907673633095680, 619593964021650475, 531571294549842067456, 559896149105493602658256, 710322778732936488128872448, 1068386732538408106621063668220, 1879866814874817967233600382304256

Offset: 0

Seiichi Manyama, May 11 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Main diagonal of A381512.

Cf. A298851, A381459.

Join[{1}, Table[Sum[(-1)^k * (n-2*k)^(3*n) * Binomial[n, k] / (2^n*n!), {k,0,n}],{n,1,20}]] (* Vaclav Kotesovec, May 13 2025 *)

a(n) = sum(k=0, n, (-1)^k*(n-2*k)^(3*n)*binomial(n, k))/(2^n*n!);

a(n) = [x^n] 1/Product_{k=0..floor(n/2)} (1 - (n-2*k)^2*x).
a(n) = (1/(2^n*n!)) * Sum_{k=0..n} (-1)^k * (n-2*k)^(3*n) * binomial(n,k).
a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.35572032955623014748562257137412853926900571707993382361... and c = 0.81034327454108346293530087910356437429774959841653144433... - Vaclav Kotesovec, May 13 2025
In closed form, a(n) ~ r^(r*n) * (1 + 2*r)^(3*n+1) * exp(n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - 8*r - 8*r^2)) * 2^(n - 1/2) * (1+r)^((1+r)*n)), where r = 0.002562299585216598238663221142585901101711497682846... is the positive real root of the equation exp(2*arctanh(1 + 2*r) - 6/(1 + 2*r)) = -1. - Vaclav Kotesovec, May 17 2025

A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1)^2 x).

Original entry on oeis.org

1, 1, 91, 24970, 14057043, 13444400190, 19558289594910, 40250341173506100, 111335096965772406915, 398473840263173643939190, 1791905773077609090895008106, 9890754761467721759394797416396, 65747198205879568307026776928408110

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Cf. A001818, A298851, A348082, A348087, A381512.

Table[SeriesCoefficient[Product[1/(1 - (2*k-1)^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

a(n) = polcoef(1/prod(k=1, n, 1-(2*k-1)^2*x+x*O(x^n)), n);

From Vaclav Kotesovec, Oct 16 2021, updated May 16 2025: (Start)
a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 52.447924272991536496097233490380538810534457762204101802471270109895148... and c = 0.028365099209561232079163758339093959048662789595134609351298413762...
In closed form, a(n) ~ 2^(2*n) * exp(2*n) * r^(n*r + 1/2) * (1+r)^(4*n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - r*(2+r))) * (2+r)^((2+r)*n - 1/2)), where r = 0.044382033760833484984013906344747760869028157215190550759633... is the root of the equation exp(4/(1+r)) = (1 + 2/r). (End)
a(n) = A381512(n,2*n-1) = (1/(2^(2*n-2)*(2*n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*n-1-2*k)^(4*n-1) * binomial(2*n-1,k) for n > 0. - Seiichi Manyama, May 16 2025

A381513 Expansion of 1/((1-x) * (1-9x) (1-25x) (1-49*x)).

Original entry on oeis.org

1, 84, 5082, 273988, 14057043, 704652312, 34924991284, 1721255653656, 84589852475205, 4151111343284620, 203559674043568206, 9978304519004079804, 489033934020664081687, 23965088084608743341808, 1174349949111469898739048, 57544663330834689436581232, 2819726398822301040064135689

Offset: 0

Seiichi Manyama, May 11 2025

Index entries for linear recurrences with constant coefficients, signature (84,-1974,12916,-11025).

Column k=7 of A381512.

a(n) = (49^(n+3)-5*25^(n+3)+9^(n+4)-5)/46080;

my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-9*x)*(1-25*x)*(1-49*x)))

E.g.f.: sinh(x)^7/7! = Sum_{k>=0} a(k) * x^(2*k+7)/(2*k+7)!.
a(n) = (49^(n+3) - 5*25^(n+3) + 9^(n+4) - 5)/46080.
a(n) = 84*a(n-1) - 1974*a(n-2) + 12916*a(n-3) - 11025*a(n-4).
a(n) = (1/645120) * Sum_{k=0..7} (-1)^k * (7-2*k)^(2*n+7) * binomial(7,k).

cp's OEIS Frontend

A348082 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1)^2 * x).

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A383837 a(n) = (3n)!/n! [x^(3*n)] sinh(x)^n.

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A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1)^2 x).

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A381513 Expansion of 1/((1-x) * (1-9x) (1-25x) (1-49*x)).

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cp's OEIS Frontend

A348082 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1)^2 * x).

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A383837 a(n) = (3*n)!/n! * [x^(3*n)] sinh(x)^n.

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A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1)^2 * x).

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A381513 Expansion of 1/((1-x) * (1-9*x) * (1-25*x) * (1-49*x)).

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PARI

Formula

A348082 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1)^2 * x).

A383837 a(n) = (3n)!/n! [x^(3*n)] sinh(x)^n.

A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1)^2 x).

A381513 Expansion of 1/((1-x) * (1-9x) (1-25x) (1-49*x)).