A327280 -id:A327280 - cp's OEIS frontend

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231

Offset: 0

Vaclav Kotesovec, Mar 25 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 31 2021

Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 25 2016

This constant is very close to exp(5*Pi/(6*sqrt(2))) / sqrt(2) = 4.502476748630924546525119125234175537729... - Vaclav Kotesovec, May 17 2018

			4.502476747617354487738593932700784406763128756091621633464540424...

Cf. A181315, A206229, A270913, A270915, A303071, A304626, A327280.

RealDigits[1/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A270913(n)^(1/n).

A327214 Self-convolution of A270913.

Original entry on oeis.org

1, 2, 7, 32, 137, 592, 2597, 11442, 50567, 224112, 995392, 4428372, 19727877, 87983202, 392755207, 1754625632, 7844003907, 35086658052, 157023432677, 703037135122, 3148915010832, 14108913792342, 63235380631747, 283495965998772, 1271282293531077, 5702105357347602

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A022567, A270913, A304443, A309986, A327215.

A270913[n_]:=SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A270913[k]*A270913[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270914 = 4.5024767476173544877385939327... and c = A327280 = 0.260542233142438469433860832160... (see A270913).

A327279 Decimal expansion of a constant related to A008485 and A327215.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 28 2019

			0.26801521271073331568695383828032867950066675724203942644590415846953909499267...

Cf. A008485, A270915, A327215, A327280, A366022.

val = Sqrt[(1 - r*s)*(Log[r*s]^2/(2*Pi*(4*ArcTanh[1 - 2*r*s]*(r*s + (-1 + r*s)*Log[r*s]) - 2*(1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] + (-1 + r*s)*(2 + 3*Log[r*s] - 2*Log[1 - r*s]) * QPolyGamma[0, 1, r*s] + (1 - r*s)* QPolyGamma[0, 1, r*s]^2 + (-1 + r*s)*(QPolyGamma[1, 1, r*s] + r*s*Log[r*s]*(r*s^2*Log[r*s] * Derivative[0, 2][QPochhammer][r*s, r*s] - 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s])))))] /. FindRoot[{QPochhammer[r*s] == 1/s, 1/s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/(s*Log[r*s])}, {r, 1/5}, {s, 2}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* Vaclav Kotesovec, Oct 02 2023 *)

Equals limit_{n->infinity} A008485(n) * sqrt(n) / A270915^n.

cp's OEIS Frontend

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327214 Self-convolution of A270913.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A327279 Decimal expansion of a constant related to A008485 and A327215.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula