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 A122920 Diagonal sums of number triangle A122919. 3
 1, 1, 4, 12, 39, 129, 436, 1498, 5218, 18386, 65420, 234734, 848403, 3086001, 11288412, 41499354, 153247278, 568188606, 2114334312, 7893906144, 29561195238, 111007927386, 417918303144, 1577061975492, 5964172347604, 22601012748124, 85806694043116, 326343785428946, 1243200250005995 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 Partial sums are Fine numbers (A000957) with offset 3. - Alexander Burstein, Apr 15 2015 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2. FORMULA G.f.: ((1-x)*(1-2*x-2*x^2-sqrt(1-4*x))/(2*(2+x)*x^3)). 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 a(n) ~ 2^(2*n+4) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 03 2014 From Vladimir Reshetnikov, Oct 26 2015: (Start) 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)!). 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) 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 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 EXAMPLE 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 + ... MATHEMATICA 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 *) 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 *) PROG (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 CROSSREFS Cf. A000957. Sequence in context: A149327 A308446 A334458 * A241073 A149328 A149329 Adjacent sequences: A122917 A122918 A122919 * A122921 A122922 A122923 KEYWORD easy,nonn AUTHOR Paul Barry, Sep 19 2006 STATUS approved

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Last modified August 6 22:48 EDT 2024. Contains 375002 sequences. (Running on oeis4.)