This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319094 #35 Dec 10 2023 18:05:20
%S A319094 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,
%T A319094 54,45,37,27,18,10,6,3,1,1,4,14,32,65,108,166,228,296,352,396,412,409,
%U A319094 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
%N 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.
%C A319094 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.
%D A319094 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
%H A319094 Shara Lalo, <a href="/A319094/a319094.pdf">Triangle of coefficients in expansions of (1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n.</a>
%H A319094 Shara Lalo, <a href="/A319094/a319094_1.pdf">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.</a>
%H A319094 Shara Lalo, <a href="/A319094/a319094_3.pdf">Formulas for the coefficients in expansions of (1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n.</a>
%F A319094 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.
%F A319094 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).
%e A319094 Triangle begins:
%e A319094 1;
%e A319094 1, 1, 2, 1, 1, 1, 1;
%e A319094 1, 2, 5, 6, 8, 8, 9, 8, 7, 4, 3, 2, 1;
%e A319094 1, 3, 9, 16, 27, 36, 47, 54, 60, 58, 54, 45, 37, 27, 18, 10, 6, 3, 1;
%e A319094 ...
%t A319094 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 A319094 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
%o A319094 (PARI) row(n) = Vecrev((1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n);
%o A319094 tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Oct 15 2018
%Y A319094 Cf. A319092, A319093, A319095.
%K A319094 tabf,nonn,easy
%O A319094 0,4
%A A319094 _Shara Lalo_, Oct 01 2018