A284860 Alternating row sums of the Sheffer triangle (exp(x), exp(3*x) - 1) given in A282629.

1, -2, -5, 19, 178, 175, -7739, -72056, -33179, 6899311, 87861076, 215532301, -11151014291, -222077806202, -1563185592617, 22953386817343, 878911293113026, 12330887396253691, 1416506544326449, -4284948239134152536

Offset: 0

Author

Wolfdieter Lang, Apr 05 2017

Keywords

• sign
• easy

Comments

See A282629 for details. This is a generalization of A000587.

Crossrefs

Cf. A000110, A282629, A284859.

Programs

Mathematica

Fold[#2 - #1 &, Reverse@ #] & /@ Table[Sum[Binomial[m, k] (-1)^(k - m) (1 + 3 k)^n/m!, {k, 0, m}], {n, 0, 19}, {m, 0, n}] (* Michael De Vlieger, Apr 08 2017 *)

PARI

T(n, m) = sum(k=0, m, binomial(m, k) * (-1)^(k - m) * (1 + 3*k)^n/m!);
a(n) = sum(m=0, n, (-1)^m*T(n, m)); \\ Indranil Ghosh, Apr 10 2017

Formula

a(n) = Sum_{m=0..n} (-1)^m*A282629(n, m), n >= 0.

E.g.f.: exp(x)*exp(1 - exp(3*x)).

a(n) = (1/e)*Sum_{m>=0} ((-1)^m / m!)*(1 + 3*m)^n, n >= 0, (Dobiński type formula).- Wolfdieter Lang, Apr 10 2017

a(0) = 1; a(n) = a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - Ilya Gutkovskiy, Nov 29 2023