cp's OEIS Frontend

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

%I A172455 #110 Feb 16 2025 08:33:11
%S A172455 1,7,84,1463,33936,990542,34938624,1445713003,68639375616,
%T A172455 3676366634402,219208706540544,14397191399702118,1032543050697424896,
%U A172455 80280469685284582812,6725557192852592984064,603931579625379293509683
%N A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.
%H A172455 Vincenzo Librandi, <a href="/A172455/b172455.txt">Table of n, a(n) for n = 1..200</a>
%H A172455 R. J. Martin and M. J. Kearney, <a href="https://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 307.
%H A172455 R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.
%H A172455 NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/9.5">Airy Functions</a>.
%H A172455 A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%H A172455 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AiryFunctions.html">Airy Functions</a>, contains the definitions of Ai(x), Bi(x).
%F A172455 a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - _Michael Somos_, Jul 24 2011
%F A172455 G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - _Michael Somos_, Jan 03 2013
%F A172455 a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [_Vladimir Reshetnikov_, Sep 24 2013]
%F A172455 a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - _Vaclav Kotesovec_, Jan 19 2015
%F A172455 6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - _Gheorghe Coserea_, May 12 2017
%F A172455 G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - _Peter Bala_, May 21 2017
%e A172455 G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
%e A172455 a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.
%t A172455 a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* _Vaclav Kotesovec_, Jan 19 2015 *)
%o A172455 (PARI) {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* _Michael Somos_, Jul 24 2011 */
%o A172455 (PARI)
%o A172455 S(v1, v2, v3, N=16) = {
%o A172455   my(a = vector(N)); a[1] = 1;
%o A172455   for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
%o A172455 };
%o A172455 S(6,-4,-1)
%o A172455 \\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
%o A172455 \\ _Gheorghe Coserea_, May 12 2017
%Y A172455 Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).
%K A172455 nonn
%O A172455 1,2
%A A172455 _N. J. A. Sloane_, Nov 20 2010