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A098278 D(n,0)/2^n, where D(n,x) is triangle A098277.

%I A098278 #45 Aug 01 2021 05:51:20
%S A098278 1,1,3,21,267,5349,154923,6120741,316271787,20701782309,1673934058923,
%T A098278 163850823271461,19093313058395307,2611858473935397669,
%U A098278 414452507370456337323,75508557963926980473381
%N A098278 D(n,0)/2^n, where D(n,x) is triangle A098277.
%C A098278 This is related to formula (1.7) in Lazar and Wachs reference.
%C A098278 Apparently all terms (except the initial 1s) have 3-valuation 1. - _F. Chapoton_, Jul 31 2021
%H A098278 Ange Bigeni and Evgeny Feigin, <a href="https://arxiv.org/abs/1808.04275">Symmetric Dellac configurations</a>, arXiv:1808.04275 [math.CO], 2018.
%H A098278 Alexander Lazar and Michelle L. Wachs, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2019/93.html">On the homogenized Linial arrangement: intersection lattice and Genocchi numbers</a>, Séminaire Lotharingien de Combinatoire, 82B.93 (FPSAC 2019).
%H A098278 A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynômes qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 1-26.
%F A098278 G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1*x/(1-1*2*x/(1-2*3*x/(1-2*4*x/...)))).
%F A098278 G.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k*(k+1)/2*x). - _Paul D. Hanna_, Sep 05 2012
%F A098278 G.f.: 1/G(0) where G(k) = 1 - x*(k+1)*(2*k+1)/(1 - x*(k+1)*(2*k+2)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jan 14 2013.
%F A098278 a(n+1) = Sum_{k=0..n} A098277(n,k)*(1/2)^k. - _Philippe Deléham_, Feb 08 2013
%e A098278 G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 267*x^4 + 5349*x^5 + ...
%e A098278 where A(x) = 1 + x/(1+x) + 2!^2*x^2/((1+x)*(1+3*x)) + 3!^2*x^3/((1+x)*(1+3*x)*(1+6*x)) + 4!^2*x^4/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) + ... - _Paul D. Hanna_, Sep 05 2012
%t A098278 d[0, _] = 1; d[n_, x_] := d[n, x] = (x+1)(x+2)d[n-1, x+2]-x(x+1)d[n-1, x];
%t A098278 a[n_] := d[n, 0]/2^n;
%t A098278 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Oct 26 2018 *)
%o A098278 (PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+k*(k+1)/2*x +x*O(x^n))), n)} \\ _Paul D. Hanna_, Sep 05 2012
%Y A098278 Cf. A000366.
%K A098278 nonn
%O A098278 0,3
%A A098278 _Ralf Stephan_, Sep 07 2004