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

1, 4, 4, 9, 24, 16, 16, 80, 128, 64, 25, 200, 560, 640, 256, 36, 420, 1792, 3456, 3072, 1024, 49, 784, 4704, 13440, 19712, 14336, 4096, 64, 1344, 10752, 42240, 90112, 106496, 65536, 16384, 81, 2160, 22176, 114048, 329472, 559104, 552960, 294912, 65536, 100

Offset: 1

Author

Michael Somos, Apr 12 2020

Keywords

• nonn
• tabl

Comments

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.

The r=4 case of the Logistic Map is 4*x*(1 - x) = -P(1, -x). The r=2 case leads to A193862.

Examples

			First four rows:
.1
.4...4
.9..24..16
16..80.128..64
		

Crossrefs

Cf. A001108, A039598, A123588, A193862.

Programs

Mathematica

T[ n_, k_] := If[k == 0, 0, Binomial[n + k - 1, 2 k - 1] 4^(k - 1) n / k];

PARI

{T(n, k) = if(k == 0, 0, binomial(n + k - 1, 2*k - 1) * 4^(k - 1) * n/k)};

Formula

P(n, x) = sinh(n * arcsinh(sqrt(x)))^2 = (hypergeom([-n, n], [1/2], -x) - 1)/2 are the row polynomials.

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)).

Row sums are A001108.

T(n, k) = (-1)^n * (-4)^(k-1) * A039598(-k - 1, n - 1) for all n in Z if k<0.

T(n, k) = -(-1)^(n+k) * A123588(n,k) if 1 <= k <= n.