A365130
G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^2)^3.
Original entry on oeis.org
1, 3, 18, 124, 945, 7650, 64592, 562419, 5013645, 45530725, 419735784, 3917714430, 36949853641, 351597275136, 3371317098546, 32542166997655, 315962469096855, 3083729075615055, 30236064140642514, 297698542934231016, 2942082095638037148
Offset: 0
-
a(n, s=2, t=3) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);
A367259
G.f. satisfies A(x) = 1 + x*A(x)^3 * (1 + x*A(x))^2.
Original entry on oeis.org
1, 1, 5, 27, 169, 1138, 8061, 59188, 446455, 3438863, 26935372, 213883631, 1717852129, 13931065117, 113913095218, 938154381748, 7774936633411, 64791892224825, 542598513709481, 4564001359135661, 38541714429405304, 326640923339410701
Offset: 0
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A367259 := proc(n)
add(binomial(3*k+(n-k)+1,k) * binomial(2*k,n-k) / (3*k+(n-k)+1),k=0..n) ;
end proc:
seq(A367259(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
-
a(n, s=2, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));
A378733
G.f. A(x) satisfies A(x) = 1 + x / (1 - x*A(x)^2)^4.
Original entry on oeis.org
1, 1, 4, 18, 96, 551, 3332, 20906, 134820, 888151, 5951096, 40432550, 277892604, 1928668910, 13497833600, 95150192558, 674993798716, 4815149310441, 34519885929860, 248571425473698, 1797058507267104, 13038781500215352, 94914559729835580, 692987915940266152
Offset: 0
-
A378733 := proc(n)
add(binomial(2*(n-k)+1, k)*binomial(n+3*k-1, n-k)/(2*(n-k)+1),k=0..n) ;
end proc:
seq(A378733(n),n=0..80) ; # R. J. Mathar, Dec 15 2024
-
a(n, r=1, s=4, t=0, u=2) = 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));