A319094 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.

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, 1, 4, 14, 32, 65, 108, 166, 228, 296, 352, 396, 412, 409, 380, 336, 276, 215, 156, 108, 68, 39, 20, 10, 4, 1, 1, 5, 20, 55, 130, 256, 455, 725, 1075, 1475, 1907, 2310, 2655, 2885

Offset: 0

Author

Shara Lalo, Oct 01 2018

Keywords

• tabf
• nonn
• easy

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.

Examples

			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;
...
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Links

• Shara Lalo, Triangle of coefficients in expansions of (1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n.
• Shara Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + x + 2 x^2 + x^3 + x^4 + x^5 + x^6)^n.
• Shara Lalo, Formulas for the coefficients in expansions of (1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n.

Crossrefs

Cf. A319092, A319093, A319095.

Programs

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

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

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).