A366221
G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^2*A(x)^3.
Original entry on oeis.org
1, 1, 5, 25, 145, 905, 5941, 40433, 282721, 2018897, 14661349, 107945993, 803922289, 6045458905, 45840518933, 350100674785, 2690717983169, 20794719218593, 161502488175557, 1259855507859193, 9867012143508305, 77554946281194793, 611575725258403061
Offset: 0
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nmax = 22; A[_] = 1;
Do[A[x_] = 1 + x*(1 + x)^2*A[x]^3 + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
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a(n) = sum(k=0, n, binomial(2*k, n-k)*binomial(3*k, k)/(2*k+1));
A382918
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^2 )^2.
Original entry on oeis.org
1, 2, 11, 64, 401, 2652, 18241, 129216, 936469, 6911238, 51764834, 392494366, 3006851913, 23238830982, 180974578418, 1418728452902, 11186978492689, 88668723061112, 706042492550773, 5645331629000370, 45307653034905824, 364860349786846894, 2947299389835541583
Offset: 0
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a(n, r=2, s=2, t=3, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));
A367280
G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x)^3)^2.
Original entry on oeis.org
1, 1, 5, 33, 251, 2073, 18069, 163600, 1523731, 14504988, 140499307, 1380322749, 13721269995, 137758098052, 1394840743638, 14227181658075, 146048314214619, 1507739540085350, 15643456882376418, 163036276218805231, 1706021256401103673
Offset: 0
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a(n, s=2, t=3, u=3) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));
Showing 1-3 of 3 results.