A365120
G.f. A(x) satisfies A(x) = (1 + x / (1 - x*A(x))^2)^2.
Original entry on oeis.org
1, 2, 5, 18, 70, 294, 1291, 5864, 27314, 129766, 626367, 3063096, 15143562, 75563924, 380062186, 1924840480, 9807649900, 50241194250, 258597717591, 1336730670244, 6936403057274, 36119232561000, 188677598254078, 988464846388710, 5192270327405662
Offset: 0
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a(n, s=2, t=2) = sum(k=0, n, binomial(t*(n-k+1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));
A367237
G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^2)^2.
Original entry on oeis.org
1, 1, 4, 20, 114, 702, 4550, 30585, 211270, 1490561, 10695354, 77809481, 572608270, 4254996670, 31882486314, 240620654468, 1827464108766, 13956516915303, 107114560278680, 825727777034002, 6390721805005678, 49638977802126104, 386824024893533450
Offset: 0
-
CoefficientList[Series[Root[-1 + #1 + x*#1^2 - 2*x*#1^3 - x^2*#1^4 + x^2*#1^5&, 1],{x,0,20}],x] (* Vaclav Kotesovec, Nov 13 2023 *)
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a(n, s=2, t=2, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));
A367238
G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x)^2)^2.
Original entry on oeis.org
1, 1, 5, 31, 219, 1672, 13439, 112043, 960017, 8402085, 74791408, 675033956, 6163120105, 56820187321, 528231686315, 4946304326883, 46609889424547, 441664236745594, 4205848369345681, 40228631544942031, 386317524696654392, 3723196299965400616
Offset: 0
-
a(n, s=2, t=3, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, 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
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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
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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));
Showing 1-4 of 4 results.