A124061
Multiplicative encoding of Catalan's triangle: Product p(i+1)^T(n,i).
Original entry on oeis.org
2, 6, 450, 2836181250, 81492043057751910481759423160156250, 4561157026363824997482074305569280581505536351717093893927260661169357729871499327113563125890139588096951624677718591308593750
Offset: 1
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 * 7^5 = 2836181250.
a(5) = 2^1 * 3^4 * 5^9 * 7^14 * 11^14 = 81492043057751910481759423160156250.
a(6) = 2^1 * 3^5 * 5^14 * 7^28 * 11^42 * 13^42.
A153046
Multiplicative encoding of Losanitsch's triangle (A034851).
Original entry on oeis.org
2, 6, 30, 3150, 6063750, 1717605545906250, 2623719141408662719128738281250, 1019408754706474658106933474548666805595768826381331909476074218750
Offset: 0
The fourth row of Losanitsch's triangle is 1, 2, 4, 2, 1 and the first five primes are 2, 3, 5, 7, 11, therefore the fourth term is 2^1 * 3^2 * 5^4 * 7^2 * 11^1 = 6063750.
Cf.
A007188 (multiplicative encoding of Pascal's triangle).
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a[n_, 0] := 1; a[n_, n_] := 1; a[n_, k_] := a[n, k] = a[n - 1, k - 1] + a[n - 1, k] - Binomial[n/2 - 1, (k - 1)/2]Mod[k, 2]Mod[n - 1, 2]; (* The above comes from Weisstein's Mathematica notebook *) multEncLoz[n_] := Times @@ Table[Prime[k + 1]^a[n, k], {k, 0, n}]; Table[multEncLoz[n], {n, 0, 7}]
A123257
Multiplicative encoding of nim sum triangle: Product p(i+1)^BitXOR(n,i).
Original entry on oeis.org
2, 6, 100, 9261000, 103306896, 16274381169926880, 98925457477919384169000000, 8078021071852487276180833326494285813758890000000, 20381485968895666256747501044033896769440000
Offset: 1
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.
A123261
Multiplicative encoding of Motzkin triangle (A026300).
Original entry on oeis.org
2, 6, 450, 405168750, 10326560651880195445980468750, 17149769349660883198128523550890723880659651223306378240865271303752564539222570800781250
Offset: 1
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.
A123558
Multiplicative encoding of the (1,2)-Pascal triangle (or Lucas triangle) A029635.
Original entry on oeis.org
2, 18, 1350, 24806250, 94588417300781250, 117849324069921797604001373181152343750, 527608657124852026883960737403192593816085584608183988660956158325750160217285156250
Offset: 1
a(1) = 2^1 = 2.
a(2) = 2^1 * 3^2 = 18.
a(3) = 2^1 * 3^3 * 5^2 = 1350.
a(4) = 2^1 * 3^4 * 5^5 * 7^2 = 24806250.
a(5) = 2^1 * 3^5 * 5^9 * 7^7 * 11^2 = 94588417300781250.
a(6) = 2^1 * 3^6 * 5^14 * 7^16 * 11^9 * 13^2 = 117849324069921797604001373181152343750.
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