A306347 Expansion of e.g.f. exp((sin(x) + sinh(x))/2).

1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 389, 1904, 9329, 38040, 132147, 542648, 3283633, 20997824, 114657097, 536178880, 2784500161, 19876061312, 153326461311, 1034551839872, 6051063485481, 38079448046208, 312420426154893, 2785055242928768, 22141255520251313

Offset: 0

Author

Ilya Gutkovskiy, May 08 2019

Keywords

• nonn

Comments

Number of partitions of n-set into blocks congruent to 1 mod 4.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..608

Crossrefs

Cf. A002017, A003724, A005046, A013025, A016813, A035451, A291975, A307978, A307979, A351969.

Programs

Mathematica

nmax = 28; CoefficientList[Series[Exp[(Sin[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 28}]

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+sinh(x))/2))) \\ Seiichi Manyama, Mar 17 2022

PARI

a(n) = if(n==0, 1, sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ Seiichi Manyama, Mar 17 2022

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(n-4*k-1). - Seiichi Manyama, Mar 17 2022