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A154344 Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.

%I A154344 #18 Jul 18 2025 09:45:04
%S A154344 1,0,-2,0,-3,3,0,-4,12,0,0,-5,35,0,-30,0,-6,90,0,-360,180,0,-7,217,0,
%T A154344 -2730,3150,-630,0,-8,504,0,-16800,33600,-15120,0,0,-9,1143,0,-91854,
%U A154344 283500,-215460,0,22680,0,-10,2550,0,-466200,2085300,-2381400,0,907200,-226800
%N A154344 Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.
%C A154344 The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=-1 multiplied by n+1 this results in a decomposition of 2^n times the Genocchi numbers A036968.
%H A154344 G. C. Greubel, <a href="/A154344/b154344.txt">Table of n, a(n) for n = 0..1274</a>
%H A154344 Ayse Yilmaz Ceylan and Yilmaz Simsek, <a href="https://doi.org/10.3390/sym17071126">Formulae for Generalization of Touchard Polynomials with Their Generating Functions</a>, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
%H A154344 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>
%F A154344 Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
%F A154344 G(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*(n+1)*c(k)*v^n.
%F A154344 A036968(n) = (1/2^n)*Sum_{k=0..n} G(n,k).
%e A154344 Triangle begins:
%e A154344   1,
%e A154344   0, -2,
%e A154344   0, -3,    3,
%e A154344   0, -4,   12, 0,
%e A154344   0, -5,   35, 0,    -30,
%e A154344   0, -6,   90, 0,   -360,    180,
%e A154344   0, -7,  217, 0,  -2730,   3150,    -630,
%e A154344   0, -8,  504, 0, -16800,  33600,  -15120, 0,
%e A154344   0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680,
%e A154344   ...
%p A154344 G := proc(n, k) local v, c; c := m -> if irem(m+1, 4) = 0 then 0 else 1/((-1)^iquo(m+1, 4)*2^iquo(m, 2)) fi; add((-1)^v*binomial(k, v)*(n+1)*c(k)*v^n, v=0..k) end: seq(print(seq(G(n, k), k=0..n)), n=0..8);
%t A154344 g[n_, k_] := Module[{v, c, pow}, pow[a_, b_] := If[ a == 0 && b == 0, 1, a^b]; c[m_] := If[ Mod[m+1, 4] == 0 , 0 , 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; Sum[(-1)^v*Binomial[k, v]*(n+1)*c[k]*pow[v, n], {v, 0, k}]]; Table[g[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 23 2013, translated from Maple *)
%Y A154344 Cf. A036968, A153641, A154341, A154342, A154343, A154345.
%K A154344 easy,sign,tabl
%O A154344 0,3
%A A154344 _Peter Luschny_, Jan 07 2009