OFFSET
0,4
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.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
LINKS
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.
G.f.: 1/(1 - t*x - t*x^2 - 2*t*x^3 - t*x^4 - t*x^5 - t*x^6 - t*x^7).
EXAMPLE
Triangle begins:
1;
1, 1, 2, 1, 1, 1, 1;
1, 2, 5, 6, 8, 8, 9, 8, 7, 4, 3, 2, 1;
1, 3, 9, 16, 27, 36, 47, 54, 60, 58, 54, 45, 37, 27, 18, 10, 6, 3, 1;
...
MATHEMATICA
Clear[t, n, k]; t[n_, k_] := t[n, k] = Sum[(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!), {i, 0, k}, {j, 2*i, k}, {q, 3*i, k}, {r, 4*i, k}, {p, 5*i, k}]; Flatten[Table[t[n, k], {n, 0, 3}, {k, 0, 6*n}]]
t[n_, k_] := t[n, k] = Sum[(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!), {i, 0, k}, {j, 0, k}, {q, 0, k}, {r, 0, k}, {p, 0, k}]; Table[t[n, k], {n, 0, 3}, {k, 0, 6*n}] // Flatten
PROG
(PARI) row(n) = Vecrev((1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
Shara Lalo, Oct 01 2018
STATUS
approved