A123261 Multiplicative encoding of Motzkin triangle (A026300).

2, 6, 450, 405168750, 10326560651880195445980468750, 17149769349660883198128523550890723880659651223306378240865271303752564539222570800781250

Offset: 1

Author

Jonathan Vos Post, Nov 06 2006

Keywords

• easy
• nonn

Comments

This is to A026300 "Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1)" as A007188 "Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i)" is to A007318 "Pascal's triangle read by rows."

Examples

			a(1) = p(1)^T(1,1) = 2^1 = 2.
a(2) = p(1)^T(2,1) * p(2)^T(2,2) = 2^1 * 3^1 = 6.
a(3) = p(1)^T(3,1) * p(2)^T(3,2) * p(3)^T(3,3) = 2^1 * 3^2 * 5^2 = 450.
a(4) = 2^1 * 3^3 * 5^5 * 7^4 = 405168750.
a(5) = 2^1 * 3^4 * 5^9 * 7^12 * 11^9 = 10326560651880195445980468750.
a(6) = 2^1 * 3^5 * 5^14 * 7^25 * 11^30 * 13^21.
a(7) = 2^1 * 3^6 * 5^20 * 7^44 * 11^69 * 13^76 * 17^51.
		

Crossrefs

Cf. A000040, A007188, A007318, A009766, A124061, Motzkin numbers (A001006) are T(n, n), other columns of T include A002026, A005322, A005323.

Formula

a(n) = Product_{i=1..n} p(i+1)^T(n,i), where T(n,i), are as in Motzkin triangle (A026300), T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).