A371398
Expansion of (1/x) * Series_Reversion( x / ( (1+x) * (1+2*x)^3 ) ).
Original entry on oeis.org
1, 7, 67, 741, 8909, 113107, 1492103, 20251945, 280978681, 3967031839, 56811348235, 823250855181, 12049087175493, 177857857845675, 2644773866954255, 39581787842355409, 595745692419162737, 9011736489133233463, 136932249972928786387
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+2*x)^3))/x)
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a(n) = sum(k=0, n, 2^k*binomial(3*(n+1), k)*binomial(n+1, n-k))/(n+1);
A108440
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and having k u=(2,1) steps among the steps leading to the first d step.
Original entry on oeis.org
1, 1, 1, 5, 4, 1, 33, 25, 7, 1, 249, 184, 54, 10, 1, 2033, 1481, 446, 92, 13, 1, 17485, 12620, 3863, 846, 139, 16, 1, 156033, 111889, 34637, 7881, 1411, 195, 19, 1, 1431281, 1021424, 318812, 74492, 14102, 2168, 260, 22, 1, 13412193, 9536113, 2995228
Offset: 0
T(2,1)=4 because we have udud, udUdd, uUddd and Uuddd.
Triangle begins:
.1;
.1,1;
.5,4,1;
.33,25,7,1;
.249,184,54,10,1;
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A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-t*z*A-z*A^2): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
`if`(x=0, 1, b(x-1, y-1, false)+b(x-1, y+2, t)+
b(x-2, y+1, t)*`if`(t, z, 1))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, true)):
seq(T(n), n=0..10); # Alois P. Heinz, Oct 06 2015
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b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, False] + b[x - 1, y + 2, t] + b[x - 2, y + 1, t]*If[t, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][ b[3*n, 0, True]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)
A371404
Expansion of (1/x) * Series_Reversion( x / ( (1+x) * (1+3*x)^2 ) ).
Original entry on oeis.org
1, 7, 64, 667, 7513, 89092, 1095832, 13852195, 178855075, 2348744095, 31273438804, 421224534100, 5728966150924, 78569975545432, 1085350298162608, 15087689038165555, 210907141968410071, 2962825568825439349, 41806163408065511032, 592244891188614804643
Offset: 0
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seq(simplify(hypergeom([-n, -2*(n+1)], [2], 3)), n = 0..20); # Peter Bala, Sep 08 2024
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+3*x)^2))/x)
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a(n) = sum(k=0, n, 3^k*binomial(2*(n+1), k)*binomial(n+1, n-k))/(n+1);