A319095 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) + 3*T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) + T(n-1, k-7) for k = 0..7*n; T(n,k) = 0 for n or k < 0.

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 3, 8, 9, 10, 15, 12, 11, 10, 9, 4, 3, 2, 1, 1, 3, 6, 16, 27, 39, 64, 78, 90, 108, 108, 102, 94, 84, 60, 46, 33, 21, 10, 6, 3, 1, 1, 4, 10, 28, 59, 104, 188, 288, 401, 556, 686, 796, 899, 944, 928, 880, 803, 668, 542, 420, 305, 200, 132, 80, 43, 20, 10, 4, 1

Offset

0,5

Comments

Row n gives the coefficients in the expansion of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, where n is a nonnegative integer. The row sum at row n is s(n) = 10^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 - x^3 - 3*x^4 - x^5 - x^6 - x^7 - x^8) 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 - 3*x^5 - x^6 - x^7 - x^8), see links.

Note: Coefficients in expansion of (1 + x + ... + x^7)^n is given in A171890 (Octonomial coefficient array).

References

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

Links

Table of n, a(n) for n=0..74.

Shara Lalo, Triangle of coefficients in expansions of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n.

Shara Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n.

Shara Lalo, Formulas for coefficients in expansions of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n.

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} Sum_{d=6*i..k}(f) for k=0..7*n; f= (3^(q - 2*r + p)*n!)/((n + d - k)!*(k + p - 2*d)!*(d + r - 2*p)!*(q + p - 2*r)!*(j + r - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!); f=0 for (n + d - k)<0 or (k + p - 2*d)<0 or (d + r - 2*p)<0 or (q + p - 2*r)<0 or (j + r - 2*q)<0 or (i + q - 2*j)<0 or (j - 2*i)<0. A novel formula proven by Shara Lalo and Zagros Lalo. Also see formula in Links section.

G.f.: 1/(1 - t*x - t*x^2 - t*x^3 - 3*t*x^4 - t*x^5 - t*x^6 - t*x^7 - t*x^8).

Example

Triangle begins:
1;
1, 1, 1,  3,  1,  1,  1,  1;
1, 2, 3,  8,  9, 10, 15, 12, 11,  10,   9,   4,  3,  2,  1;
1, 3, 6, 16, 27, 39, 64, 78, 90, 108, 108, 102, 94, 84, 60, 46, 33, 21, 10, 6, 3, 1;
...

Mathematica

f[n_] := CoefficientList[(1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, x] ; Join @@ Table[f[k], {k, 0, 5}] // Flatten (* Amiram Eldar, Dec 07 2018 *)

Prog

(PARI) row(n) = Vecrev((1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018
(Maxima) T(n, k) := ratcoef((1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, x, k)$
create_list(T(n, k), n, 0, 10, k, 0, 7*n); /* Franck Maminirina Ramaharo, Nov 27 2018 */

Crossrefs

Cf. A171890.

Sequence in context: A066636 A137578 A200146 * A109390 A130046 A349509

Adjacent sequences: A319092 A319093 A319094 * A319096 A319097 A319098

Keyword

tabf,nonn,easy

Author

Shara Lalo, Oct 01 2018

Status

approved