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));
A366216
G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^4.
Original entry on oeis.org
1, 1, 8, 60, 520, 4886, 48384, 497460, 5259872, 56834345, 624819148, 6966612604, 78592083420, 895432704860, 10288759392156, 119089472755860, 1387274092156508, 16251727492295812, 191342076640423136, 2262894045516407118, 26869820052175649836
Offset: 0
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a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(4*k, k)/(3*k+1));
A366222
G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^3.
Original entry on oeis.org
1, 1, 7, 42, 287, 2114, 16338, 130802, 1075355, 9025656, 77021482, 666267502, 5829209046, 51492030953, 458612500526, 4113879873624, 37133888342707, 337041718357465, 3074153880004188, 28162578841220534, 259020296989987934, 2390818256963083305
Offset: 0
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a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(3*k, k)/(2*k+1));
A379037
G.f. A(x) satisfies A(x) = ( (1 + x) * (1 + x*A(x)^(3/2)) )^2.
Original entry on oeis.org
1, 4, 18, 106, 689, 4782, 34707, 260190, 1999168, 15660176, 124596498, 1004110948, 8179379807, 67239070868, 557098881919, 4647368670950, 39001655222787, 329048378867468, 2789241880512898, 23743798316713368, 202894843070927859, 1739775692700850554
Offset: 0
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a(n) = 2*sum(k=0, n, binomial(3*k+2, k)*binomial(3*k+2, n-k)/(3*k+2));
A382894
G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^2.
Original entry on oeis.org
1, 2, 13, 78, 520, 3664, 26859, 202808, 1566693, 12323982, 98381841, 795023284, 6490951398, 53462144788, 443683640945, 3706539244272, 31144893093298, 263052053436600, 2231992880546400, 19016760502183968, 162629329186013523, 1395500273826639540
Offset: 0
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a(n, r=2, s=3, t=3, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));
Showing 1-5 of 5 results.