A364475
G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^3.
Original entry on oeis.org
1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 34304706, 230923653, 1569684910, 10759159000, 74281473504, 516089542684, 3605685460750, 25316226436086, 178538289189108, 1264131169628799, 8982889404251721, 64041351551534215
Offset: 0
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A364475 := proc(n)
add( binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k)/(2*n-2*k+1),k=0..n/2) ;
end proc:
seq(A364475(n),n=0..80); # R. J. Mathar, Jul 27 2023
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a(n) = sum(k=0, n\2, binomial(3*n-3*k, k)*binomial(3*n-4*k, n-2*k)/(2*n-2*k+1));
A137265
G.f. y(x) is solution of x y^3 - (1 + x^2) y + 1 = 0 with y(0) = 1.
Original entry on oeis.org
1, 1, 2, 8, 35, 163, 796, 4024, 20885, 110654, 596064, 3254752, 17974893, 100227022, 563482140, 3190633232, 18179765509, 104158703503, 599698459613, 3467978715612, 20134256546896, 117313279477959, 685756774642494, 4020515276730588, 23636036336651811
Offset: 0
a(3) = 8 because g(x) = 1 + x + 2 x^2 + 8 x^3 + O(x^4) satisfies x*g(x)^3 - (1 + x^2)*g(x) + 1 = O(x^4).
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f:= (x,y) -> x*y^3 - (1 + x^2)*y + 1; N:= (y,n) -> convert(normal(taylor(y-f(x,y)/D[2](f)(x,y),x=0,n)),polynom); Y:= 1; for j from 1 to 6 do Y:= N(Y,2^j) end do; seq(coeftayl(Y,x=0,j),j=0..2^6-1);
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max = 22; g[x_] := Sum[a[k]*x^k, {k, 0, max}]; coes = CoefficientList[ Series[ x*g[x]^3 - (1+x^2)*g[x] + 1, {x, 0, max}], x]; sol = First[ Solve[ Thread[ coes == 0 ] ] ]; Table[a[n] /. sol, {n, 0, max}](* Jean-François Alcover, Nov 28 2011 *)
terms = 25; y[] = 1; Do[y[x] = (1 + x*y[x]^3)/(1 + x^2) + O[x]^terms, terms]; CoefficientList[y[x], x] (* Jean-François Alcover, Jan 11 2018 *)
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a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1)); \\ Seiichi Manyama, Nov 02 2023
A364478
G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^8.
Original entry on oeis.org
1, 1, 4, 23, 154, 1124, 8675, 69626, 575243, 4859778, 41789764, 364565277, 3218581695, 28702642553, 258172627259, 2339496034381, 21337716782873, 195726876816623, 1804472496834650, 16711389876481027, 155395461519245354, 1450298253483719944
Offset: 0
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a(n) = sum(k=0, n\2, binomial(3*n+2*k, k)*binomial(3*n+k, n-2*k)/(2*n+3*k+1));
A367040
G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.
Original entry on oeis.org
1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025
Offset: 0
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a(n) = sum(k=0, n\2, binomial(2*(n-2*k)+1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));
A364473
G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^6.
Original entry on oeis.org
1, 1, 3, 13, 65, 353, 2024, 12057, 73890, 462851, 2950261, 19073921, 124776881, 824409052, 5493384031, 36874564529, 249114808794, 1692489908494, 11556616157589, 79265016880139, 545860966841247, 3772800724433931, 26162662010039826, 181974370638420829
Offset: 0
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a(n) = sum(k=0, n\2, binomial(2*n+2*k, k)*binomial(2*n+k, n-2*k)/(n+3*k+1));
Showing 1-5 of 5 results.