A361638
Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^2 * (1 + A(x)^3).
Original entry on oeis.org
1, 2, 14, 142, 1690, 21994, 303126, 4348102, 64235570, 970695442, 14934154334, 233133082494, 3683546302538, 58794776161274, 946619511627622, 15355445768326710, 250717346336174690, 4117189670041072930, 67956239699290313646, 1126763233375565370990
Offset: 0
-
a(n) = sum(k=0, n, binomial(n, k)*binomial(2*n+3*k+1, n)/(2*n+3*k+1));
A371657
G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x) + A(x)^2).
Original entry on oeis.org
1, 3, 27, 333, 4752, 73764, 1209492, 20610693, 361403937, 6478386561, 118181952369, 2186908154748, 40949739595242, 774474351098031, 14772979729013247, 283878381945510621, 5490264493926636912, 106786725176131118523, 2087502569999563971843
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 3^(n-k)*binomial(n, k)*binomial(3*n-k, n-1-2*k))/n);
A378155
G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(2/3) * (1 + A(x)^(2/3)) )^3.
Original entry on oeis.org
1, 6, 48, 452, 4680, 51504, 591312, 7002864, 84926304, 1049402944, 13165069824, 167239042176, 2146912312064, 27808372643328, 362981425115904, 4769884412086016, 63050983340533248, 837805424714425344, 11184489029495865344, 149935005483457542144, 2017560365768892739584
Offset: 0
-
a(n, r=3, t=2, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));
A378156
G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(1/2) * (1 + A(x)^(1/2)) )^4.
Original entry on oeis.org
1, 8, 72, 720, 7728, 87104, 1017184, 12200640, 149429504, 1861059328, 23498407680, 300110580224, 3870135336192, 50323754919936, 659085377250816, 8686436702866432, 115120162870534144, 1533214282017157120, 20510220228874399744, 275462154992599851008, 3712900128220039372800
Offset: 0
-
a(n, r=4, t=2, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));