A319093 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) + 2*T(n-1, k-3) + T(n-1, k-4) for k = 0..4*n; T(n,k)=0 for n or k < 0.
1, 1, 1, -1, 2, 1, 1, 2, -1, 2, 7, -2, 2, 4, 1, 1, 3, 0, 1, 15, 3, -4, 24, 6, -1, 9, 6, 1, 1, 4, 2, 0, 23, 20, -14, 48, 55, -24, 46, 52, 2, 12, 20, 8, 1, 1, 5, 5, 0, 30, 51, -15, 60, 180, -25, 49, 280, 15, 30, 180, 72, 15, 45, 35, 10, 1, 1, 6, 9, 2, 36, 96, 11, 54, 387, 116, -51, 774, 376, -162, 804, 532
Offset: 0
Examples
Triangle begins: 1; 1, 1, -1, 2, 1; 1, 2, -1, 2, 7, -2, 2, 4, 1; 1, 3, 0, 1, 15, 3, -4, 24, 6, -1, 9, 6, 1; 1, 4, 2, 0, 23, 20, -14, 48, 55, -24, 46, 52, 2, 12, 20, 8, 1; 1, 5, 5, 0, 30, 51, -15, 60, 180, -25, 49, 280, 15, 30, 180, 72, 15, 45, 35, 10, 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 - x^2 + 2*x^3 + x^4)^n.
- Shara Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + x - x^2 + 2*x^3 + x^4)^n.
- Shara Lalo, Formulas for Coefficients in Expansion of (1 + x - x^2 + 2*x^3 + x^4)^n.
Crossrefs
Cf. A319096.
Programs
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Mathematica
Clear[t, n, k]; t[n_, k_] := t[n, k] = Sum[((-1)^(i + q - 2*j)*2^(j - 2*i)*n!)/((n - k + q)!*(k + j - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!), {i, 0, k}, {j, 2*i, k}, {q, 3*i, k}]; Flatten[Table[t[n, k], {n, 0, 7}, {k, 0, 4*n}]] Clear[t, n, k]; t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + t[n - 1, k - 1] - t[n - 1, k - 2] + 2 t[n - 1, k - 3] + t[n - 1, k - 4]]; Table[t[n, k], {n, 0, 6}, {k, 0, 4*n} ] // Flatten -
PARI
row(n) = Vecrev((1 + x - x^2 + 2*x^3 + x^4)^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}(f) for k = 0..4*n; f= (-1)^(i + q - 2*j)*2^(j - 2*i)*n!)/((n - k + q)!*(k + j - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!); f=0 for (n - k + q)<0 or (k + j - 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 - 2*t*x^4 - t*x^5).
Comments