A051029 Ramanujan's b-series: expansion of (2-26x-12x^2)/(1-82x-82x^2+x^3).
2, 138, 11468, 951690, 78978818, 6554290188, 543927106802, 45139395574362, 3746025905565260, 310875010766342202, 25798879867700837522, 2140996154008403172108, 177676881902829762447458, 14745040201780861879966890, 1223660659865908706274804428
Offset: 0
References
- For additional references and links see A051028.
Links
- Robert Israel, Table of n, a(n) for n = 0..468
- Kwang-Wu Chen, Extensions of an amazing identity of Ramanujan, Fib. Q., 50 (2012), 227-230.
- Jung Hun Han and Michael D. Hirschhorn, Another Look at an Amazing Identity of Ramanujan, Mathematics Magazine, Vol. 79 (2006), pp. 302-304.
- Michael D. Hirschhorn, An amazing identity of Ramanujan, Math. Mag. 68 (1995), no. 3, 199--201. MR1335148
- Michael D. Hirschhorn, A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan, Math. Mag., 69.4 (1996), 267-269.
- J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
- Eric Rowland and Jesus Sistos Barron, Complexity of powers of a constant-recursive sequence, arXiv:2501.14643 [math.NT], 2025. See p. 2.
- Eric Weisstein's World of Mathematics, Ramanujan's Sum Identity.
- Index entries for linear recurrences with constant coefficients, signature (82,82,-1).
Programs
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Maple
g:=(2-26*x-12*x^2)/(1-82*x-82*x^2+x^3): gser:=series(g,x=0,20): seq(coeff(gser,x,n),n=0..12); # Emeric Deutsch, Oct 14 2006
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Mathematica
CoefficientList[Series[(2 - 26 x - 12 x^2)/(1 - 82 x - 82 x^2 + x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 22 2015 *) -
PARI
Vec((2-26*x-12*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ Michel Marcus, Feb 29 2016
Formula
G.f.: (2-26*x-12*x^2)/((1+x)*(1-83*x+x^2)).
X(n+1) = A*X(n), where X(n) = transpose(A051028(n), A051029(n), A051030(n)) and A = matrix(3,3,[63,104,-68; 64,104,-67; 80,131,-85]). - Emeric Deutsch, Oct 14 2006
Extensions
Minor edits (g.f. and name) by M. F. Hasler, May 08 2016
Comments