A344920 The Worpitzky transform of the squares.

0, -1, 5, -13, 29, -61, 125, -253, 509, -1021, 2045, -4093, 8189, -16381, 32765, -65533, 131069, -262141, 524285, -1048573, 2097149, -4194301, 8388605, -16777213, 33554429, -67108861, 134217725, -268435453, 536870909, -1073741821, 2147483645, -4294967293

Offset: 0

Author

Peter Luschny, Jun 24 2021

Keywords

• sign
• easy

Comments

The Worpitzky transform maps a sequence A to a sequence B, where B(n) = Sum_{k=0..n} A163626(n, k)*A(k). (If A(n) = 1/(n + 1) then B(n) are the Bernoulli numbers (with B(1) = 1/2.))

Also row 2 in A371761. Can be generated by the signed Akiyama-Tanigawa algorithm for powers (see the Python script). - Peter Luschny, Apr 12 2024

Links

• Index entries for linear recurrences with constant coefficients, signature (-3,-2).

Crossrefs

Up to shift and sign: even bisection A267921, odd bisection A141725.

Cf. A163626, A036563, A371761.

Programs

Maple

gf := (exp(x) - 1)*(exp(x) - 2)*exp(-2*x): ser := series(gf, x, 36):
seq(n!*coeff(ser, x, n), n = 0..31);

Mathematica

W[n_, k_] := (-1)^k k! StirlingS2[n + 1, k + 1];
WT[a_, len_] := Table[Sum[W[n, k] a[k], {k, 0, n}], {n, 0, len-1}];
WT[#^2 &, 32] (* The Worpitzky transform applied to the squares. *)

Python

# Using the Akiyama-Tanigawa algorithm for powers from A371761.
print([(-1)**n * v for (n, v) in enumerate(ATPowList(2, 32))])
# Peter Luschny, Apr 12 2024

Formula

a(n) = n! * [x^n] (exp(x) - 1)*(exp(x) - 2)*exp(-2*x).

a(n) = (-1)^(n + 1)*(3 - 2^(n + 1)) for n >= 1. - Hugo Pfoertner, Jun 24 2021

a(n) = [x^n] x*(2*x - 1)/(2*x^2 + 3*x + 1). - Stefano Spezia, Jun 24 2021