The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A143556 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^3. 6
 1, 1, 6, 18, 110, 498, 3366, 17282, 122958, 672930, 4938758, 28103730, 210595182, 1230391058, 9358456230, 55727128866, 428643977422, 2589488117826, 20092671283974, 122759098980690, 959216278565742, 5913900861617970 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..450 FORMULA G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)). G.f. satisfies: A(x) = 1 + x^2 + x*A(x)^3/A(-x)^2. G.f. satisfies: (A(x) - 1)^2 = ( 1 - (1+x^2)/A(x) )^3/x = x^2*A(x)^6/A(-x)^6. G.f.: A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^3/G(-x)^2 is the g.f. of A143562. G.f. satisfies: x*A(x)^5 - 2*x*A(x)^4 - (1-x)*A(x)^3 + 3*(1+x^2)*A(x)^2 - 3*(1+x^2)^2*A(x) + (1+x^2)^3 = 0. Recurrence: 4*(n-1)*n*(2*n-5)*(2*n+1)*(2916*n^10 - 99630*n^9 + 1494855*n^8 - 12945798*n^7 + 71493183*n^6 - 262308129*n^5 + 645244282*n^4 - 1046448887*n^3 + 1066283852*n^2 - 614660500*n + 152638416)*a(n) = 60*(n-1)*(13122*n^11 - 458217*n^10 + 7044759*n^9 - 62741439*n^8 + 358008636*n^7 - 1365100815*n^6 + 3513825159*n^5 - 6010387373*n^4 + 6521940316*n^3 - 4078695988*n^2 + 1207261712*n - 113170176)*a(n-1) + 15*(n-2)*(160380*n^13 - 6121170*n^12 + 104460435*n^11 - 1051745310*n^10 + 6938544798*n^9 - 31476010053*n^8 + 100128993299*n^7 - 223244300184*n^6 + 341877397736*n^5 - 343306364591*n^4 + 206330136024*n^3 - 62025904772*n^2 + 8101283136*n - 2665897920)*a(n-2) + 450*(n-4)*(7020*n^10 - 107820*n^9 + 91377*n^8 + 9009842*n^7 - 87380558*n^6 + 404731832*n^5 - 1079876519*n^4 + 1690685386*n^3 - 1439622136*n^2 + 509372600*n + 4226320)*a(n-3) + 750*(n-5)*(n-4)*(14580*n^12 - 498150*n^11 + 7512345*n^10 - 65844630*n^9 + 371440818*n^8 - 1409248026*n^7 + 3643384398*n^6 - 6348642805*n^5 + 7178246227*n^4 - 4869145209*n^3 + 1716210104*n^2 - 292182404*n + 75613440)*a(n-4) + 3750*(n-6)*(n-5)*(n-4)*(3240*n^8 - 52785*n^7 + 324459*n^6 - 854916*n^5 + 387102*n^4 + 2695803*n^3 - 5239793*n^2 + 2713946*n + 268800)*a(n-5) + 3125*(n-7)*(n-6)*(n-5)*(n-4)*(2916*n^10 - 70470*n^9 + 729405*n^8 - 4223718*n^7 + 14971977*n^6 - 33317457*n^5 + 45697282*n^4 - 36099439*n^3 + 14258060*n^2 - 2573132*n + 694560)*a(n-6). - Vaclav Kotesovec, Mar 25 2014 a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n), where {r1 = r = 0.13384151194121538538097804723..., s1 = 1.57588974374012701113388456...} and {r2 = -r, s2 = 0.9688941320566492403600185...} are roots of the system of equations r*(r^5 + 3*r*(s-1)^2 + (s-1)^2*s^3) = 3*r^4*(s-1) + (s-1)^3, r*(s-1)*(6*r + s^2*(5*s-3)) = 3*(r^4 + (s-1)^2), and c = c1+c2 = 0.525673619703566161096484... if n is even, and c = c1-c2 = 0.471796676012154625609556... if n is odd, where c1 = M(r1,s1), c2=M(r2,s2), and M(r,s) = sqrt(r*(6*r^5 - 12*r^3*(s-1) + 6*r*(s-1)^2 + (s-1)^2*s^3)/(3+3*r^2-3*s+r*s*(3-12*s+10*s^2)))/2. - Vaclav Kotesovec, Mar 25 2014 EXAMPLE G.f. A(x) = 1 + x + 6*x^2 + 18*x^3 + 110*x^4 + 498*x^5 + 3366*x^6 +... A(x)/A(-x) = 1 + 2*x + 2*x^2 + 26*x^3 + 50*x^4 + 706*x^5 + 1650*x^6 +... A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 60*x^3 + 208*x^4 + 1716*x^5 +... where 1 - (1+x^2)/A(x) = x*A(x)^2/A(-x)^2. PROG (PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^3/subst(A^3, x, -x)); polcoeff(A, n)} CROSSREFS Cf. A143562, A143555, A143557, A143558, A143559. Sequence in context: A367664 A361729 A108735 * A007126 A009576 A009580 Adjacent sequences: A143553 A143554 A143555 * A143557 A143558 A143559 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 24 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 21 16:52 EDT 2024. Contains 372738 sequences. (Running on oeis4.)