A285065 Alternating row sums of Sheffer triangle S2[4,1] = A285061.

1, -3, -7, 53, 497, -147, -44055, -437339, 971745, 90858205, 1254551513, -56188139, -361749699119, -7793811482035, -47717641321527, 2053219888651909, 77548473901557697, 1171383881442334141, -8155337883596701767

Offset: 0

Author

Wolfdieter Lang, Apr 13 2017

Keywords

• sign
• easy

Comments

See A285061 for details. This is a generalization of A000587.

Crossrefs

Cf. A000587, A285061, A285064.

Programs

Mathematica

Table[Sum[Binomial[n, k]*BellB[k, -1]*4^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2017 *)

Python

from sympy import bell, binomial
def a(n): return sum([binomial(n, k)*bell(k, -1)*4**k for k in range(n + 1)]) # Indranil Ghosh, May 06 2017

Formula

a(n) = Sum_{m=0..n} (-1)^m*A285061(n, m), n >= 0.

E.g.f.: exp(x)*exp(1 - exp(4*x)).

a(n) = e*Sum_{m>=0} ((-1)^m / m!)*(1 + 4*m)^n, n >= 0, (Dobiński type formula).

a(n) = Sum_{k=0..n} binomial(n, k) * 4^k * A000587(k), n >= 0. - Vaclav Kotesovec, Apr 23 2017

a(0) = 1; a(n) = a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k). - Ilya Gutkovskiy, Nov 30 2023