A364924
G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - 2*x*A(x)^4).
Original entry on oeis.org
1, 1, 7, 67, 743, 8970, 114445, 1517976, 20722023, 289224355, 4108588558, 59207805442, 863439906413, 12718638581368, 188960182480440, 2828238875318256, 42605850936335463, 645497106959662857, 9829072480785776101, 150345303724987825021
Offset: 0
-
a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));
A378668
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)^2) )^2.
Original entry on oeis.org
1, 2, 13, 112, 1104, 11778, 132374, 1543740, 18505996, 226632616, 2823110349, 35659080952, 455652487060, 5879489288828, 76502741016012, 1002670573618324, 13224761472453756, 175404372357915096, 2338003752387818372, 31302169754776944512, 420760252068869028028
Offset: 0
-
a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+2, n)/(4*n+k+2));
-
a(n, r=2, s=1, t=5, u=4) = 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));
A378669
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(4/3)/(1 - x*A(x)^(4/3)) )^3.
Original entry on oeis.org
1, 3, 21, 187, 1878, 20277, 229806, 2696523, 32478204, 399230972, 4988220669, 63165060093, 808828667104, 10455471983550, 136255868388684, 1788233397919211, 23614059664575324, 313531617379965156, 4183068478829324388, 56052027108881747724, 754020313029799707018
Offset: 0
-
a(n) = 3*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+3, n)/(4*n+k+3));
-
a(n, r=3, s=1, t=5, u=4) = 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));