A367016
G.f. satisfies A(x) = 1 + x*A(x)^4*(1 - x).
Original entry on oeis.org
1, 1, 3, 14, 78, 475, 3057, 20446, 140702, 989789, 7085635, 51451482, 378049810, 2805616460, 20999408480, 158337719608, 1201585477436, 9170328295222, 70339328959266, 541953619822062, 4192560258116202, 32552250308843605, 253583917423039079
Offset: 0
-
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k, n-k)*binomial(4*k, k)/(3*k+1));
A378786
G.f. A(x) satisfies A(x) = 1 + x * (1+x)^2 * A(x)^4.
Original entry on oeis.org
1, 1, 6, 39, 296, 2435, 21138, 190603, 1767968, 16761424, 161697576, 1582171216, 15664531716, 156637712953, 1579664567130, 16048129755157, 164085811289360, 1687224436103842, 17436287104620980, 181001686332329224, 1886522317836670988, 19734386503541838083
Offset: 0
-
a(n, r=1, s=2, t=4, 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));
A381937
G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.
Original entry on oeis.org
1, 2, 6, 35, 240, 1805, 14386, 119365, 1020136, 8918423, 79380514, 716911887, 6553219720, 60513355786, 563648995020, 5289485238552, 49963186247220, 474655663418546, 4532279676629700, 43473774550929628, 418706702628897708, 4047555977981218963
Offset: 0
-
a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(k+1, n-k)/(4*k+1));
A382885
G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x) * A(x) )^3.
Original entry on oeis.org
1, 3, 18, 121, 900, 7110, 58598, 498153, 4336533, 38463732, 346368351, 3158325168, 29102914959, 270582713670, 2535191045652, 23913087584045, 226892934532149, 2164080724942155, 20737076963936828, 199542537271568802, 1927347504059464995, 18679645863925666721
Offset: 0
-
a(n, r=3, s=1, t=4, 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));