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A334009 Triangle read by rows: T(n, k) = binomial(n + k - 1, 2*k - 1) * 4^(k - 1) * n/k, 1 <= k <= n.

%I A334009 #3 Apr 12 2020 11:36:33
%S A334009 1,4,4,9,24,16,16,80,128,64,25,200,560,640,256,36,420,1792,3456,3072,
%T A334009 1024,49,784,4704,13440,19712,14336,4096,64,1344,10752,42240,90112,
%U A334009 106496,65536,16384,81,2160,22176,114048,329472,559104,552960,294912,65536,100
%N A334009 Triangle read by rows: T(n, k) = binomial(n + k - 1, 2*k - 1) * 4^(k - 1) * n/k,  1 <= k <= n.
%C A334009 Let P(n, x) := Sum_{k=1..n} T(n, k)*x^k. Then P(n, P(m, x)) = P(n*m, x) for all n and m in Z.
%C A334009 The r=4 case of the Logistic Map is 4*x*(1 - x) = -P(1, -x). The r=2 case leads to A193862.
%F A334009 P(n, x) = sinh(n * arcsinh(sqrt(x)))^2 = (hypergeom([-n, n], [1/2], -x) - 1)/2 are the row polynomials.
%F A334009 G.f.: Sum_{n, m} T(n, k) * x^k * y^n = x * y * (1 + y) / ((1 - y) * (1 - (2 + 4*x)*y + y^2)).
%F A334009 Row sums are A001108.
%F A334009 T(n, k) = (-1)^n * (-4)^(k-1) * A039598(-k - 1, n - 1) for all n in Z if k<0.
%F A334009 T(n, k) = -(-1)^(n+k) * A123588(n,k) if 1 <= k <= n.
%e A334009 First four rows:
%e A334009 .1
%e A334009 .4...4
%e A334009 .9..24..16
%e A334009 16..80.128..64
%t A334009 T[ n_, k_] := If[k == 0, 0, Binomial[n + k - 1, 2 k - 1] 4^(k - 1) n / k];
%o A334009 (PARI) {T(n, k) = if(k == 0, 0, binomial(n + k - 1, 2*k - 1) * 4^(k - 1) * n/k)};
%Y A334009 Cf. A001108, A039598, A123588, A193862.
%K A334009 nonn,tabl
%O A334009 1,2
%A A334009 _Michael Somos_, Apr 12 2020