A098279 a(n) = D(n,1)/2^n, where D(n,x) is triangle A098277.
1, 2, 10, 98, 1594, 38834, 1323658, 60134210, 3511695322, 256306614866, 22861774551466, 2446866564603362, 309483997093321210, 45666236465616727538, 7774748058886412485834
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 10*x^2 + 98*x^3 + 1594*x^4 + 38834*x^5 +... where A(x) = 1 + 2!*x/(1+x) + 2!*3!*x^2/((1+x)*(1+3*x)) + 3!*4!*x^3/((1+x)*(1+3*x)*(1+6*x)) + 4!*5!*x^4/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) + ... - Paul D. Hanna, Sep 05 2012
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..250
- Ange Bigeni, Enumerating the symplectic Dellac configurations, arXiv:1705.03804 [math.CO], 2017.
- Ange Bigeni, Evgeny Feigin, Poincaré polynomials of the degenerate flag varieties of type C, arXiv:1804.10804 [math.CO], 2018.
- Ange Bigeni, Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
- Xin Fang and Ghislain Fourier, Torus fixed points in Schubert varieties and Genocchi numbers, arXiv:1504.03980 [math.RT], 2015.
- A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.
Programs
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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, 1]/2^n; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 27 2018 *) -
PARI
{a(n)=polcoeff(sum(m=0, n, m!*(m+1)!*x^m/prod(k=1, m, 1+k*(k+1)/2*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 05 2012
Formula
G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*2*x/(1-1*3*x/(1-2*4*x/(1-2*5*x/...)))).
G.f.: Sum_{n>=0} n!*(n+1)! * x^n / Product_{k=1..n} (1 + k*(k+1)/2*x). - Paul D. Hanna, Sep 05 2012
G.f.: 2*G(0) - 1 where G(k) = 1 + x*(2*k+1)*(4*k+1)/( 1 + x + 6*x*k + 8*x*k^2 - 2*x*(k+1)*(4*k+3)*(1 + x + 6*x*k + 8*x*k^2)/(2*x*(k+1)*(4*k+3) + (1 + 6*x + 14*x*k + 8*x*k^2)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 11 2013
a(n) ~ 2^(3*n+11/2) * n^(2*n+2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Apr 23 2015