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A098278 D(n,0)/2^n, where D(n,x) is triangle A098277. 2
1, 1, 3, 21, 267, 5349, 154923, 6120741, 316271787, 20701782309, 1673934058923, 163850823271461, 19093313058395307, 2611858473935397669, 414452507370456337323, 75508557963926980473381 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Ralf Stephan, Sep 07 2004
STATUS
approved

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Last modified March 3 08:35 EST 2024. Contains 370499 sequences. (Running on oeis4.)