A124061 -id:A124061 - cp's OEIS frontend

A123257 Multiplicative encoding of nim sum triangle: Product p(i+1)^BitXOR(n,i).

2, 6, 100, 9261000, 103306896, 16274381169926880, 98925457477919384169000000, 8078021071852487276180833326494285813758890000000, 20381485968895666256747501044033896769440000

Offset: 1

Jonathan Vos Post, Nov 06 2006

This is to A003987 "Table of n XOR m (or Nim-sum of n and m)" as A007188 "Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i)" is to A007318 "Pascal's triangle read by rows." T[2i,2j] = 2T[i,j], T[2i+1,2j] = 2T[i,j] + 1. a(2^n-1) = (n#)^(2^n-1) = A002110(n)^A000225(n).

			a(1) = p(1)^T(1,1) = 2^1 = 2, where T(i,j) is as in A003987.
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) = 2^2 * 3^0 * 5^2 = 100.
a(4) = 2^3 * 3^3 * 5^3 * 7^3 = 9261000.
a(5) = 2^4 * 3^2 * 5^0 * 7^2 * 11^4 = 103306896.
a(6) = 2^5 * 3^5 * 5^1 * 7^1 * 11^5 * 13^5 = 16274381169926880.
a(7) = 2^6 * 3^4 * 5^6 * 7^0 * 11^6 * 13^4 * 17^6 = 98925457477919384169000000.
a(8) = 2^7 * 3^7 * 5^7 * 7^7 * 11^7 * 13^7 * 17^7 * 19^7.
a(9) = 2^8 * 3^6 * 5^4 * 7^6 * 11^0 * 13^6 * 17^4 * 19^6 * 23^8.
a(10) = 2^9 * 3^9 * 5^5 * 7^5 * 11^1 * 13^1 * 17^5 * 19^5 * 23^9 * 29^9.

Cf. A000040, A000225, A002110, A003987, A007188, A007318, A009766, A124061.

a(n) = Prod[i=i..n] p(i+1)^BitXOR(n,i).

A123261 Multiplicative encoding of Motzkin triangle (A026300).

Original entry on oeis.org

2, 6, 450, 405168750, 10326560651880195445980468750, 17149769349660883198128523550890723880659651223306378240865271303752564539222570800781250

Offset: 1

Jonathan Vos Post, Nov 06 2006

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."

			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.

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

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).

cp's OEIS Frontend

A123257 Multiplicative encoding of nim sum triangle: Product p(i+1)^BitXOR(n,i).

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