[seqfan] Pascal's triangles
юрий герасимов
2stepan at rambler.ru
Fri Nov 2 17:31:15 CET 2012
Dear Seg Fan, Dear Neil J. A. S.,
Pascal's triangle has higher dimensional generalizations of the plan.
Example:
The q-nomial arrays (n-th row sum is equal to q^n or (x + 2) ^ 2) are for q = 2..10 or x = 0..8 (x = q - 2): A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171889, A213652, A213651.
If x called index of extension, y is index of asymmetry, z is index of obliquely, the other exemles are:
1) the twonomial arrays (q = 2 or x = 0) for y = 0..4 (for all z = 0): A007318 (Pascal), A140998, A140994, A140996, A141020;
2) the twonomial arrays (q = 2 or x = 0) for y = 0..4 (for all z = 0): A007318 (Pascal), A140993, A140997, A140995, A141021.
But then why there are no freenomial arrays (q = 3 or x = 1) for y = 1..4 (for all z = 0) ?
But then why there are no treenomial arrays (q = 3 or x = 1) for y = 1..4 (for all z = 1) ?
But then why there are no quadrinomial arrays (q = 4 or x = 2) for y = 1..4 (for all z = 0) ?
But then why there are no quadrinomial arrays (q = 4 or x = 2) for y = 1..4 (for all z = 1) ?
But then why there are no 5-nomial.. e. t. c.
For a positive answer to the questions and for z of valus 0 or 1 I have proposed to use the following initial conditions 1)..3) and recurence formula 4):
1) G(n, k) = 0 for 0 > k > x*n, n >= 0, x > 0:
2) G(n, x*z*n) = G(n+1, i - z + a*z - a*i + a*x - a*x*z) = G (n+2, x*n - x*z*n + x - x*z) = 1 if y = 1, then a = 0, if y > 1, then a = 1, 1 <= i <= x;
3) G(n + r + 1, x*r - x + i - 2x*z*r + 4x*z - 2z*i + 2x*z*n) = 2*G(r, j + 2x*z - 2z*j + x*z*n) = 2*(x+1)^(r - 1) for 1 <= r <= (y - 1) and 1 <= j <= (x - 1);
4) G(n + y + 1, k) = x^(y - 1) *Sum {i..x} (G(n + 1, k - i + 2z*i - x*y*z )) + Sum {m..y} (x^(y - m)*G(n + m, k - x*y*z + x*z*m)) for 1 <= m <=y,
if z = 0, then 2 <= k <= (x*n - x*y + 3x - 1) and if z = 1, then (x*y - x + 1) <= k <= (x*n + 2x - 1).
Example:
Freenomial array with x = 1 (q = 3), y = 1, z = 1 begins: 1
1...1...1
1...2...2...3...1
1...2...2...6...8...7..1
1..
Freenomial array with x = 1 (q = 3), y = 2, z = 1 begins: 1
1...1...1
1...2...2...3...1
1...2...2 6...6...9...1
1...2...2...6...6...18..22..23..1
1...2...2...6...6...21..27,,,57,,58..1
Freenomial array with x = 1 (q = 3), y = 3, z = 1 begins: 1
1...1...1
1...2 2...3...1
1...2...2...6...6...9...1
1...2...2...6...6...18..18...27..1
1...2...2...6...6...18..18..54..62..73..1
Quadrinomial array with x = 2 (q = 4), y = 1, z = 1 begins: 1
1...1...1...1
1...2...2...3...3...4...1
1...
I am very surprised it's not already in the OEIS. Under what name and in what form to offer for publication in the OEIS ?
Tranks is advance.
Juri-Stepan Gerasimov.
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