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%I #36 Jul 08 2017 13:00:40
%S 1,1,4,12,39,129,436,1498,5218,18386,65420,234734,848403,3086001,
%T 11288412,41499354,153247278,568188606,2114334312,7893906144,
%U 29561195238,111007927386,417918303144,1577061975492,5964172347604,22601012748124,85806694043116,326343785428946,1243200250005995
%N Diagonal sums of number triangle A122919.
%C Starting with offset 1 = iterates of M * [1,1,1,0,0,0,...] where M is the tridiagonal matrix with [0,2,2,2,...] as the main diagonal and [1,1,1,...] as the super and subdiagonals. - _Gary W. Adamson_, Jan 09 2009
%C Partial sums are Fine numbers (A000957) with offset 3. - _Alexander Burstein_, Apr 15 2015
%H G. C. Greubel, <a href="/A122920/b122920.txt">Table of n, a(n) for n = 0..1000</a>
%H Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
%F G.f.: ((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3)).
%F Conjecture: 2*n*(n+3)*a(n) - (7*n^2+9*n+4)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2) = 0. - _R. J. Mathar_, Nov 05 2012
%F a(n) ~ 2^(2*n+4) / (3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 03 2014
%F From _Vladimir Reshetnikov_, Oct 26 2015: (Start)
%F a(n) = 9/(16*(-2)^n) + 3*(2*n+4)!*hypergeom([1,n+5/2,n+3], [n+2,n+5], -8)/((n+1)!*(n+4)!).
%F a(n) = 9/(16*(-2)^n) + 8*2^n*(2*n+5)!!*hypergeom([1,n+7/2], [n+5], -8)/(n+4)! - 4*2^n*(2*n+3)!!*hypergeom([1,n+5/2], [n+4], -8)/(n+3)!. (End)
%F G.f. A(x) =: y satisfies 0 = (1 - x)^2 - y*(1 - 3*x + 2*x^3) + y^2*(2*x^3 + x^4). - _Michael Somos_, Oct 26 2015
%F 0 = a(n)*(+16*a(n+1) - 26*a(n+2) - 98*a(n+3) + 36*a(n+4)) + a(n+1)*(+50*a(n+1) + 35*a(n+2) - 179*a(n+3) + 46*a(n+4)) + a(n+2)*(+105*a(n+2) + 47*a(n+3) - 50*a(n+4)) + a(n+3)*(+14*a(n+3) + 4*a(n+4)) for all n>=0. - _Michael Somos_, Oct 26 2015
%e G.f. = 1 + x + 4*x^2 + 12*x^3 + 39*x^4 + 129*x^5 + 436*x^6 + 1498*x^7 + 5218*x^8 + ...
%t CoefficientList[Series[((1-x)*(1-2*x-2*x^2-Sqrt[1-4*x])/(2*(2+x)*x^3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 03 2014 *)
%t Table[9/(16 (-2)^n) + 3 (2n+4)! HypergeometricPFQ[{1, n+5/2, n+3}, {n+2, n+5}, -8]/((n+1)! (n+4)!), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 26 2015 *)
%o (PARI) x='x+O('x^66); Vec(((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3))) \\ _Joerg Arndt_, May 08 2013
%Y Cf. A000957.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 19 2006
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