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

1, 1, 3, 21, 267, 5349, 154923, 6120741, 316271787, 20701782309, 1673934058923, 163850823271461, 19093313058395307, 2611858473935397669, 414452507370456337323, 75508557963926980473381

Offset

0,3

Comments

This is related to formula (1.7) in Lazar and Wachs reference.

Apparently all terms (except the initial 1s) have 3-valuation 1. - F. Chapoton, Jul 31 2021

Links

Table of n, a(n) for n=0..15.

Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.

Alexander Lazar and Michelle L. Wachs, On the homogenized Linial arrangement: intersection lattice and Genocchi numbers, Séminaire Lotharingien de Combinatoire, 82B.93 (FPSAC 2019).

A. Randrianarivony and J. Zeng, Une famille de polynômes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Formula

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/...)))).

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

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.

a(n+1) = Sum_{k=0..n} A098277(n,k)*(1/2)^k. - Philippe Deléham, Feb 08 2013

Example

G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 267*x^4 + 5349*x^5 + ...
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

Mathematica

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];
a[n_] := d[n, 0]/2^n;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 26 2018 *)

Prog

(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

Crossrefs

Cf. A000366.

Sequence in context: A012131 A322224 A221094 * A269938 A277454 A215127

Adjacent sequences: A098275 A098276 A098277 * A098279 A098280 A098281

Keyword

nonn

Author

Ralf Stephan, Sep 07 2004

Status

approved