cp's OEIS Frontend

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.

%I A319095 #35 Dec 10 2023 18:01:57
%S A319095 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,
%T A319095 64,78,90,108,108,102,94,84,60,46,33,21,10,6,3,1,1,4,10,28,59,104,188,
%U A319095 288,401,556,686,796,899,944,928,880,803,668,542,420,305,200,132,80,43,20,10,4,1
%N 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.
%C A319095 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.
%C A319095 Note: Coefficients in expansion of (1 + x + ... + x^7)^n is given in A171890 (Octonomial coefficient array).
%D A319095 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
%H A319095 Shara Lalo, <a href="/A319095/a319095_3.pdf">Triangle of coefficients in expansions of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n.</a>
%H A319095 Shara Lalo, <a href="/A319095/a319095_4.pdf">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.</a>
%H A319095 Shara Lalo, <a href="/A319095/a319095_6.pdf">Formulas for coefficients in expansions of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n.</a>
%F A319095 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.
%F A319095 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).
%e A319095 Triangle begins:
%e A319095 1;
%e A319095 1, 1, 1,  3,  1,  1,  1,  1;
%e A319095 1, 2, 3,  8,  9, 10, 15, 12, 11,  10,   9,   4,  3,  2,  1;
%e A319095 1, 3, 6, 16, 27, 39, 64, 78, 90, 108, 108, 102, 94, 84, 60, 46, 33, 21, 10, 6, 3, 1;
%e A319095 ...
%t A319095 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 *)
%o A319095 (PARI) row(n) = Vecrev((1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n);
%o A319095 tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Oct 15 2018
%o A319095 (Maxima) T(n, k) := ratcoef((1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, x, k)$
%o A319095 create_list(T(n,k), n, 0, 10, k, 0, 7*n); /* _Franck Maminirina Ramaharo_, Nov 27 2018 */
%Y A319095 Cf. A171890.
%K A319095 tabf,nonn,easy
%O A319095 0,5
%A A319095 _Shara Lalo_, Oct 01 2018