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A344920 The Worpitzky transform of the squares.

%I A344920 #13 Apr 12 2024 13:47:57
%S A344920 0,-1,5,-13,29,-61,125,-253,509,-1021,2045,-4093,8189,-16381,32765,
%T A344920 -65533,131069,-262141,524285,-1048573,2097149,-4194301,8388605,
%U A344920 -16777213,33554429,-67108861,134217725,-268435453,536870909,-1073741821,2147483645,-4294967293
%N A344920 The Worpitzky transform of the squares.
%C A344920 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 A344920 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 A344920 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2).
%F A344920 a(n) = n! * [x^n] (exp(x) - 1)*(exp(x) - 2)*exp(-2*x).
%F A344920 a(n) = (-1)^(n + 1)*(3 - 2^(n + 1)) for n >= 1. - _Hugo Pfoertner_, Jun 24 2021
%F A344920 a(n) = [x^n] x*(2*x - 1)/(2*x^2 + 3*x + 1). - _Stefano Spezia_, Jun 24 2021
%p A344920 gf := (exp(x) - 1)*(exp(x) - 2)*exp(-2*x): ser := series(gf, x, 36):
%p A344920 seq(n!*coeff(ser, x, n), n = 0..31);
%t A344920 W[n_, k_] := (-1)^k k! StirlingS2[n + 1, k + 1];
%t A344920 WT[a_, len_] := Table[Sum[W[n, k] a[k], {k, 0, n}], {n, 0, len-1}];
%t A344920 WT[#^2 &, 32] (* The Worpitzky transform applied to the squares. *)
%o A344920 (Python)
%o A344920 # Using the Akiyama-Tanigawa algorithm for powers from A371761.
%o A344920 print([(-1)**n * v for (n, v) in enumerate(ATPowList(2, 32))])
%o A344920 # _Peter Luschny_, Apr 12 2024
%Y A344920 Up to shift and sign: even bisection A267921, odd bisection A141725.
%Y A344920 Cf. A163626, A036563, A371761.
%K A344920 sign,easy
%O A344920 0,3
%A A344920 _Peter Luschny_, Jun 24 2021