A381867
G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x)^2, where C(x) is the g.f. of A000108.
Original entry on oeis.org
1, 3, 10, 44, 239, 1464, 9610, 65946, 466951, 3385259, 24999475, 187385168, 1421901090, 10901237530, 84312106160, 657031204068, 5153954345309, 40663760712441, 322478148002872, 2569086552458460, 20551321340065924, 165009872444132477, 1329352163579556971, 10742386009423170696
Offset: 0
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a(n) = sum(k=0, n, binomial(3*k+1, k)*binomial(n+k+1, n-k)/(3*k+1));
A254747
a(n) = (1 + Sum_{j=0..n} (C(n,j)*C(3*j-1,j))) / 2.
Original entry on oeis.org
1, 2, 8, 47, 312, 2162, 15311, 109965, 797824, 5833298, 42910998, 317224800, 2354712927, 17538747124, 131017428431, 981194304302, 7364370502896, 55380344444150, 417176211054422, 3147365470080480, 23777750075552262
Offset: 0
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a := n -> (hypergeom([1/3, 2/3, -n], [1/2, 1], -27/4) +2 ) / 3:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 07 2015
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FullSimplify[CoefficientList[Series[1 + x*D[Log[(2*Sin[(1/3)* ArcSin[(3/2)*Sqrt[3]* Sqrt[x/(1 - x)]]])/ (Sqrt[3]*Sqrt[(1 - x)* x])], x], {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 07 2015 *)
Table[(1 + Sum[Binomial[n, j]*Binomial[3*j-1, j], {j, 0, n}])/2, {n,0,20}] (* Vaclav Kotesovec, Feb 07 2015 after Vladimir Kruchinin *)
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a(n):=(1+sum(binomial(n,j)*binomial(3*j-1,j),j,0,n))/2;
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for(n=0,25, print1((1 + sum(k=0,n, binomial(n,k)*binomial(3*k-1,k)))/2, ", ")) \\ G. C. Greubel, Jun 03 2017
A346763
G.f. A(x) satisfies: A(x) = 1 / (1 - 3*x) + x * (1 - 3*x) * A(x)^3.
Original entry on oeis.org
1, 4, 18, 93, 550, 3636, 26079, 197931, 1562382, 12685116, 105187512, 886700898, 7574331987, 65413265014, 570155069547, 5008957733472, 44306834969838, 394269180748272, 3527034255411864, 31700659283908242, 286124960854479888, 2592334353741781752, 23567790327842864046
Offset: 0
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nmax = 22; A[] = 0; Do[A[x] = 1/(1 - 3 x) + x (1 - 3 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
Table[3^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -9/4], {n, 0, 22}]
A381829
G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.
Original entry on oeis.org
1, 2, 12, 97, 905, 9187, 98578, 1099980, 12636101, 148449436, 1775331503, 21541303494, 264533752068, 3281596216087, 41062196808517, 517655936768189, 6568539787903369, 83827401412072474, 1075254139150601581, 13855040994605807348, 179256835556387995412, 2327788724156294034612
Offset: 0
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a(n) = sum(k=0, n, binomial(3*n+1, k)*binomial(4*n-3*k, n-k))/(3*n+1);
A381832
G.f. A(x) satisfies A(x) = C(x*A(x)^3) / (1 - x), where C(x) is the g.f. of A000108.
Original entry on oeis.org
1, 2, 10, 81, 796, 8616, 98973, 1184324, 14602486, 184219731, 2366543116, 30851212416, 407106050261, 5427274340091, 72986372975716, 988937692146346, 13487903251385562, 185022817888443780, 2551096865411701371, 35335463473311506321, 491444773227779518956, 6860346682881319595632
Offset: 0
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a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(n+2*k, n-k)/(5*k+1));
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