A333042 -id:A333042 - cp's OEIS frontend

A229451 G.f.: exp( Sum_{n>=1} (3n)!/n!^3 x^n/n ).

1, 6, 63, 866, 13899, 246366, 4676768, 93322596, 1934035965, 41286407510, 902562584556, 20119266633060, 455832458083577, 10470568749165246, 243361203186769659, 5714294570067499930, 135377464019074334826, 3232534121305720233264, 77726654423445817800164

Offset: 0

Paul D. Hanna, Sep 23 2013

The sixth root of the o.g.f. A(x)^(1/6) = 1 + x + 8*x^2 + 101*x^3 + 1569*x^4 + 27445*x^5 + ... appears to have integer coefficients. See A229452. More generally, if A(m,x) := exp( Sum_{n >= 1} (m*n)!/n!^m * x^n/n ), m = 1,2,3,..., then it can be shown that the expansion of A(m,x) has integer coefficients. We conjecture that the expansion of A(m,x)^(1/m!) also has integer coefficients. - Peter Bala, Feb 16 2020

			G.f.: A(x) = 1 + 6*x + 63*x^2 + 866*x^3 + 13899*x^4 + 246366*x^5 +...,
where
log(A(x)) = 6*x + 90*x^2/2 + 1680*x^3/3 + 34650*x^4/4 + 756756*x^5/5 +...+ A006480(n)*x^n/n + ....

Seiichi Manyama, Table of n, a(n) for n = 0..702

Cf. A229452, A006480 (de Bruijn's S(3,n)), A061401, A333042, A333043, A370288, A362732, A370289, A370293.

CoefficientList[Series[Exp[6*x*HypergeometricPFQ[{1,1,4/3,5/3},{2,2,2},27*x]],{x,0,20}],x] (* Vaclav Kotesovec, Dec 25 2013 *)

{a(n)=polcoeff(exp(sum(k=1,n,(3*k)!/k!^3*x^k/k) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ c * 3^(3*n)/n^2, where c = 2^11 * 3^(7/2) * Pi^5 * A370293^6 = 0.406436497... - Vaclav Kotesovec, Dec 25 2013, updated Feb 14 2024
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A006480(k) * a(n-k). - Seiichi Manyama, Feb 09 2024
From Peter Bala, Oct 24 2024: (Start)
Series reversion of x*A(-x) = x + 6*x^2 + 9*x^3 + 56*x^4 - 300*x^5 + 3942*x^6 - ... is the g.f. of A061401.
The g.f. A(x) satisfies [x^n] A(x)^n = A362732(n). (End)

A370294 G.f.: exp(Sum_{k>=1} (4k)!/(4!k!^4) * x^k/k).

Original entry on oeis.org

1, 1, 53, 5186, 663444, 98703235, 16179000550, 2837251240021, 522937525075783, 100134345595461824, 19762585810520535829, 3997199042964419204924, 825055790810846248226675, 173231819660726985218760834, 36906136513918240767383588700, 7962139696794640558535530147729

Offset: 0

Vaclav Kotesovec, Feb 14 2024

nonn

Cf. A008977, A082368, A229452, A370295, A333042.

CoefficientList[Series[Exp[Sum[(4*k)!/(4!*k!^4)*x^k/k, {k, 1, 20}]], {x, 0, 20}], x]
CoefficientList[Series[Exp[x*HypergeometricPFQ[{1, 1, 5/4, 3/2, 7/4}, {2, 2, 2, 2}, 256*x]], {x, 0, 20}], x]

G.f. A(x) = G(x)^(1/24), where G(x) is the g.f. for A333042.

a(n) ~ c * 4^(4*n)/n^(5/2), where c = exp(HypergeometricPFQ[{1, 1, 5/4, 3/2, 7/4}, {2, 2, 2, 2}, 1] / 256) / (24*sqrt(2)*Pi^(3/2)) = 0.005320414767134132512371690902604699480645296829596277834542636529157577...

A333043 G.f.: exp(Sum_{k>=1} (5k)!/k!^5 x^k/k).

Original entry on oeis.org

1, 120, 63900, 63148000, 85136103750, 137629764435024, 250331826090382280, 494436455370401985600, 1037731227148399567352625, 2281874234819846601146115000, 5205960892339635531670022801628, 12237148815599682784939438806708960, 29483782935554473122496294160376815950

Offset: 0

Vaclav Kotesovec, Mar 06 2020

nonn

In general, if r>=2, m>0 and g.f. = exp(m * Sum_{k>=1} (r*k)!/k!^r * x^k/k), then a(n) ~ c(r,m) * m * r^(r*n + 1/2) / ((2*Pi)^((r-1)/2) * n^((r+1)/2)) , where c(r,m) = exp((m * r! / r^r) * HypergeometricPFQ[{1, 1, (r+1)/r, (r+2)/r, ... , (2*r-1)/r}, {2, 2, ...r-times... 2, 2}, 1]). - Vaclav Kotesovec, Feb 16 2024

Cf. A000108, A008978, A229451, A333042, A370295.

CoefficientList[Series[Exp[Sum[(5*k)!/k!^5*x^k/k, {k, 1, 20}]], {x, 0, 20}], x]
CoefficientList[Series[Exp[120*x*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 3125*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2024 *)

a(n) ~ c * 5^(5*n)/n^3, where c = sqrt(5) * exp(24*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 1] / 625) / (4*Pi^2) = 0.05943406... - Vaclav Kotesovec, Mar 06 2020, updated Feb 16 2024

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A008978(k) * a(n-k). - Seiichi Manyama, Feb 09 2024

A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24n) = (4n)!/n!^4 for n >= 0.

Original entry on oeis.org

1, 1, 29, 2246, 239500, 30318701, 4271201506, 647359627557, 103476937050223, 17223017775652625, 2959285397777331751, 521687007046376376544, 93932798602803741121051, 17215649571517858590782737, 3203146941738318544432065500, 603763082812549420389330837978, 115095760617137117019641563685386

Offset: 0

Peter Bala, Oct 20 2024

Compare with A000984(n) = [x^n] (1 + x)^(2*n) = (2*n)!/n!^2.
The central binomial coefficients A000984(n) satisfy the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) for all primes p >= 5 and positive integers n and k.
More generally, for positive integers r and s, the sequence {u(r,s; n) : n >= 0} defined by u(r,s; n) = [x^(s*n)] (1 + x)^(r*n) = binomial(r*n, s*n) satisfies the same supercongruences (Meštrović, Section 6, equation 39).
Conjecture: for positive integers r and s, the sequence {v(r,s; n) : n >= 0} defined by v(r,s; n) = [x^(s*n)] A(x)^(r*n) also satisfies the same supercongruences.

Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

Cf. A008977, A082368, A333042, A370294, A377217, A377219.

Order := 25:
E(x) := exp(add((4*n)!/n!^4 * x^n/n, n = 1..25)):
solve(series(x*E(x),x) = y, x):
convert(%, polynom):
g := taylor(y/%, y = 0, 25):
seq(coeftayl(g^(1/24), y = 0,  n), n = 0..20);

O.g.f.: A(x) = ( x/(x * series_reversion(E(x)))^(1/24), where E(x) = exp(Sum_{n >= 1} (4*n)!/n!^4 *x^n/n) is the o.g.f. of A333042.

cp's OEIS Frontend

A229451 G.f.: exp( Sum_{n>=1} (3n)!/n!^3 x^n/n ).

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A370294 G.f.: exp(Sum_{k>=1} (4k)!/(4!k!^4) * x^k/k).

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A333043 G.f.: exp(Sum_{k>=1} (5k)!/k!^5 x^k/k).

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A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24n) = (4n)!/n!^4 for n >= 0.

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cp's OEIS Frontend

A229451 G.f.: exp( Sum_{n>=1} (3*n)!/n!^3 * x^n/n ).

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A370294 G.f.: exp(Sum_{k>=1} (4*k)!/(4!*k!^4) * x^k/k).

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A333043 G.f.: exp(Sum_{k>=1} (5*k)!/k!^5 * x^k/k).

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Formula

A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24*n) = (4*n)!/n!^4 for n >= 0.

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Formula

A229451 G.f.: exp( Sum_{n>=1} (3n)!/n!^3 x^n/n ).

A370294 G.f.: exp(Sum_{k>=1} (4k)!/(4!k!^4) * x^k/k).

A333043 G.f.: exp(Sum_{k>=1} (5k)!/k!^5 x^k/k).

A377218 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24n) = (4n)!/n!^4 for n >= 0.