A193437 -id:A193437 - cp's OEIS frontend

A193438 Expansion of e.g.f. exp( Sum_{n>=0} x^(4n+1)/(4n+1) ).

1, 1, 1, 1, 1, 25, 145, 505, 1345, 43345, 481825, 3027025, 13679425, 528618025, 8796735025, 81517983625, 529655946625, 23619691278625, 526089195906625, 6512769913326625, 55783484692170625, 2802281186570685625, 78369733286598300625, 1221751619270220585625

Offset: 0

Paul D. Hanna, Jul 25 2011

nonn

Conjecture: a(n) is divisible by 5^floor(n/5) for n>=0.

Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 4).

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + 25*x^5/5! + 145*x^6/6! + 505*x^7/7! +...
where
log(A(x)) = x + x^5/5 + x^9/9 + x^13/13 + x^17/17 + x^21/21 + x^25/25 +...

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000246, A193437.

nmax = 30; CoefficientList[Series[Exp[x*Hypergeometric2F1[1/4, 1, 5/4, x^4]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

{a(n)=n!*polcoeff( exp(sum(m=0,n,x^(4*m+1)/(4*m+1))+x*O(x^n)) ,n)}

seq(n) = Vec(serlaplace(exp(intformal(1/(1-x^4) + O(x*x^n)) ))) \\ Andrew Howroyd, Oct 15 2023

a(n) = a(n-1) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Apr 15 2020
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k)! * a(n-4*k-1). - Ilya Gutkovskiy, Jul 14 2021
E.g.f.: A(x) = exp(Integral_{z = 0..x} 1/(1-z^4) dz) = exp((arctan(x)+arctanh(x))/2). - Fabian Pereyra, Oct 14 2023

A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3k + 1) / (3k + 1)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 37, 114, 478, 1907, 6777, 28414, 148579, 758916, 3580189, 18981485, 117883917, 720627553, 4193077474, 26795418840, 191751387094, 1352954503595, 9301704998742, 69285817230370, 559142785301527, 4453089770243547, 35182348161102172

Offset: 0

Ilya Gutkovskiy, Apr 08 2020

nonn

Number of partitions of n-set into blocks congruent to 1 mod 3.

Cf. A000110, A003724, A193437, A306347, A333882, A333883.

nmax = 27; CoefficientList[Series[Exp[Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 3]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]
nmax = 30; CoefficientList[Series[Exp[Exp[x]/3 - 2*Sin[Pi/6 - Sqrt[3]*x/2] / (3*Exp[x/2])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

E.g.f.: exp(exp(x)/3 - 2*sin(Pi/6 - sqrt(3)*x/2) / (3*exp(x/2))). - Vaclav Kotesovec, Apr 15 2020

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(n-3*k-1). - Seiichi Manyama, Sep 22 2023

A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+1) / (5k+1) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 3684241, 50309281, 369738721, 1926648361, 7980936601, 1335634023361, 27705746752321, 302258931418081, 2283161710263841, 13419441405835201, 2498339829188508481, 70152448708746111961, 1025314852704395518441

Offset: 0

Seiichi Manyama, Sep 23 2023

nonn

Cf. A000246, A193437, A193438.

Cf. A333882, A365977.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\5, x^(5*k+1)/(5*k+1)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/5)} a(n-5*k-1)/(n-5*k-1)!.

a(0) = a(1) = ... = a(4) = 1; a(n) = a(n-1) + 120 * binomial(n-1,5) * a(n-5).

cp's OEIS Frontend

A193438 Expansion of e.g.f. exp( Sum_{n>=0} x^(4n+1)/(4n+1) ).

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A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3k + 1) / (3k + 1)!).

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A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+1) / (5k+1) ).

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cp's OEIS Frontend

A193438 Expansion of e.g.f. exp( Sum_{n>=0} x^(4*n+1)/(4*n+1) ).

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A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3*k + 1) / (3*k + 1)!).

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A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5*k+1) / (5*k+1) ).

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A193438 Expansion of e.g.f. exp( Sum_{n>=0} x^(4n+1)/(4n+1) ).

A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3k + 1) / (3k + 1)!).

A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+1) / (5k+1) ).