A355440 Expansion of e.g.f. Sum_{k>=0} exp(4^k * x) * x^k/k!.

1, 2, 10, 98, 2050, 84482, 7221250, 1218502658, 421846581250, 288641130823682, 403002184457781250, 1112950376623239069698, 6251793960501383945781250, 69503063309910921346390425602, 1568447691296998939150390025781250

Offset: 0

Author

Seiichi Manyama, Jul 02 2022

Keywords

• nonn

Comments

a(n) is the number of labeled digraphs on [n] with self loops allowed (A002416) such that reflexive nodes are only adjacent to irreflexive nodes and vice versa. A reflexive node is a node with a self loop. An irreflexive node is a node without a self loop. - Geoffrey Critzer, Aug 21 2023

Crossrefs

Column k=4 of A355395.

Cf. A193199.

Programs

Mathematica

nn = 6; B[n_] := n! 4^Binomial[n, 2]; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
Table[B[n], {n, 0, nn}] CoefficientList[Series[e[x]^2, {x, 0, nn}], x]

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(4^k*x)*x^k/k!)))

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-4^k*x)^(k+1)))

PARI

a(n) = sum(k=0, n, 4^(k*(n-k))*binomial(n, k));

Formula

G.f.: Sum_{k>=0} x^k/(1 - 4^k * x)^(k+1).

a(n) = Sum_{k=0..n} 4^(k*(n-k)) * binomial(n,k).

E(x)^2 = Sum_{k>=0} a(n)*x^n/B(n) where B(n) = n!*4^binomial(n,2) and E(x) = Sum_{n>=0} x^n/B(n). - Geoffrey Critzer, Aug 21 2023