STATUS
editing
approved
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Revision History for A319094
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + 2 T(n-1, k-2) + T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) for k = 0..6*n; T(n,k)=0 for n or k < 0.
(history; published version)
(history; published version)
NAME
Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + 2 T(n-1, k-2) + T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6)) for k = 0..6*n; T(n,k)=0 for n or k < 0.
PROG
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018
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approved
editing
STATUS
reviewed
approved
STATUS
proposed
reviewed
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editing
proposed
COMMENTS
Row n gives the coefficients in the expansion of (1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n, where n is a nonnegative integer. The row sum at row n is s(n) = 8^n. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in the expansion of 1/(1 - x - x^2 - 2*x^3 - x^4 - x^5 - x^6 - x^7) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in the expansion of 1/(1 - x - x^2 - x^3 - x^4 - 2*x^5 - x^6 - x^7), see links. The central coefficients are given in A319098.
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approved
editing
STATUS
proposed
approved
STATUS
editing
proposed
FORMULA
T(n,k) = Sum_{i=0..k} Sum_{j=2*i..k} Sum_{q=3*i..k} Sum_{r=4*i..k} Sum_{p=5*i..k }(f) for k=0..6*n; f = (2^(q - 2*r + p)*n!)/((n + p - k)!*(k + r - 2*p)!*(q - 2*r + p)!*(j - 2*q + r)!*(i - 2*j + q)!*(j - 2*i)!*i!); f=0 for (n + p - k)<0 or (k + r - 2*p)<0 or (q - 2*r + p)<0 or (j - 2*q + r)<0 or (i - 2*j + q)<0 or (j - 2*i)<0. A novel formula proven by Shara Lalo and Zagros Lalo. Also see formula in Links section.
STATUS
proposed
editing
STATUS
editing
proposed