A360667
Triangle read by rows: T(n,m)=4^(n-1)*C(n,m)*C(3*n/2-2,n-1)/n, for 0 <= m <= n, with T(0,0)=1.
Original entry on oeis.org
1, 1, 1, 2, 4, 2, 10, 30, 30, 10, 64, 256, 384, 256, 64, 462, 2310, 4620, 4620, 2310, 462, 3584, 21504, 53760, 71680, 53760, 21504, 3584, 29172, 204204, 612612, 1021020, 1021020, 612612, 204204, 29172, 245760, 1966080, 6881280, 13762560, 17203200, 13762560, 6881280, 1966080, 245760
Offset: 0
Triangle T(n, m) starts:
[0] 1;
[1] 1, 1;
[2] 2, 4, 2;
[3] 10, 30, 30, 10;
[4] 64, 256, 384, 256, 64;
[5] 462, 2310, 4620, 4620, 2310, 462;
[6] 3584, 21504, 53760, 71680, 53760, 21504, 3584;
[7] 29172, 204204, 612612, 1021020,1021020, 612612, 204204, 29172;
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T[0, 0] = 1;
T[n_, m_] := 4^(n-1)*Binomial[n, m]*Binomial[3n/2-2, n-1]/n;
Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 16 2023 *)
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T(n,m):=if n=0 and m=0 then 1 else if n=0 then 0 else (4^(n-1)*binomial(n,m)*binomial((3*n)/2-2,n-1))/(n);
A367384
Expansion of g.f. A(x) satisfying A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.
Original entry on oeis.org
1, 2, 16, 172, 2120, 28264, 396192, 5746480, 85394656, 1291778368, 19805198784, 306834276416, 4793670528640, 75415927948416, 1193652980090880, 18994846756882176, 303766882134726144, 4880209392051146752, 78739290124904116224, 1275444751485628848128, 20735204112205333970944
Offset: 1
G.f.: A(x) = x + 2*x^2 + 16*x^3 + 172*x^4 + 2120*x^5 + 28264*x^6 + 396192*x^7 + 5746480*x^8 + 85394656*x^9 + 1291778368*x^10 + ...
where A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 36*x^4 + 408*x^5 + 5184*x^6 + 70512*x^7 + 1002864*x^8 + 14711456*x^9 + 220670592*x^10 + ...
A(x)^3 = x^3 + 6*x^4 + 60*x^5 + 716*x^6 + 9384*x^7 + 130344*x^8 + 1882576*x^9 + 27950736*x^10 + ...
Let Ai(x) be the series reversion of A(x), then
Ai(x)^2 = A(x)^2 - 8*A(x)^3 = x^2 - 4*x^3 - 12*x^4 - 72*x^5 - 544*x^6 - 4560*x^7 - 39888*x^8 - 349152*x^9 - 2935296*x^10 - ...
and
Ai(x) = sqrt(A(x)^2 - 8*A(x)^3) = x - 2*x^2 - 8*x^3 - 52*x^4 - 408*x^5 - 3512*x^6 - 31584*x^7 - 287056*x^8 - 2560288*x^9 - ...
Also,
A(A(x)) = x + 4*x^2 + 40*x^3 + 512*x^4 + 7392*x^5 + 114688*x^6 + 1867008*x^7 + 31457280*x^8 + 543921664*x^9 + ... + 2^n*A078531(n)*x^(n+1) + ...
which satisfies A(A(x))^2 - 8*A(A(x))^3 = x^2, where
A(A(x))^2 = x^2 + 8*x^3 + 96*x^4 + 1344*x^5 + 20480*x^6 + 329472*x^7 + ...
A(A(x))^3 = x^3 + 12*x^4 + 168*x^5 + 2560*x^6 + 41184*x^7 + 688128*x^8 + ...
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{a(n) = my(A=1,V=[1]); for(i=1,n, V = concat(V,0); A = x*Ser(V);
V[#V] = polcoeff( x - subst(A,x, sqrt(A^2 - 8*A^3)), #V)/2 );V[n]}
for(n=1,30,print1(a(n),", "))