%I #13 Apr 12 2024 13:47:57

%S 0,-1,5,-13,29,-61,125,-253,509,-1021,2045,-4093,8189,-16381,32765,

%T -65533,131069,-262141,524285,-1048573,2097149,-4194301,8388605,

%U -16777213,33554429,-67108861,134217725,-268435453,536870909,-1073741821,2147483645,-4294967293

%N The Worpitzky transform of the squares.

%C 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.))

%C 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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2).

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

%F a(n) = (-1)^(n + 1)*(3 - 2^(n + 1)) for n >= 1. - _Hugo Pfoertner_, Jun 24 2021

%F a(n) = [x^n] x*(2*x - 1)/(2*x^2 + 3*x + 1). - _Stefano Spezia_, Jun 24 2021

%p gf := (exp(x) - 1)*(exp(x) - 2)*exp(-2*x): ser := series(gf, x, 36):

%p seq(n!*coeff(ser, x, n), n = 0..31);

%t W[n_, k_] := (-1)^k k! StirlingS2[n + 1, k + 1];

%t WT[a_, len_] := Table[Sum[W[n, k] a[k], {k, 0, n}], {n, 0, len-1}];

%t WT[#^2 &, 32] (* The Worpitzky transform applied to the squares. *)

%o (Python)

%o # Using the Akiyama-Tanigawa algorithm for powers from A371761.

%o print([(-1)**n * v for (n, v) in enumerate(ATPowList(2, 32))])

%o # _Peter Luschny_, Apr 12 2024

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

%Y Cf. A163626, A036563, A371761.

%K sign,easy

%O 0,3

%A _Peter Luschny_, Jun 24 2021