A304252 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 6*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 1, 1, 6, 1, 12, 1, 18, 36, 1, 24, 108, 1, 30, 216, 216, 1, 36, 360, 864, 1, 42, 540, 2160, 1296, 1, 48, 756, 4320, 6480, 1, 54, 1008, 7560, 19440, 7776, 1, 60, 1296, 12096, 45360, 46656, 1, 66, 1620, 18144, 90720, 163296, 46656, 1, 72, 1980, 25920, 163296, 435456, 326592, 1, 78, 2376, 35640

Offset: 0

Author

Zagros Lalo, May 09 2018

Keywords

• tabf
• nonn
• easy

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013613 ((1+6*x)^n).

The coefficients in the expansion of 1/(1-x-6*x^2) are given by the sequence generated by the row sums.

Examples

			Triangle begins:
  1;
  1;
  1,  6;
  1, 12;
  1, 18,   36;
  1, 24,  108;
  1, 30,  216,   216;
  1, 36,  360,   864;
  1, 42,  540,  2160,   1296;
  1, 48,  756,  4320,   6480;
  1, 54, 1008,  7560,  19440,    7776;
  1, 60, 1296, 12096,  45360,   46656;
  1, 66, 1620, 18144,  90720,  163296,   46656;
  1, 72, 1980, 25920, 163296,  435456,  326592;
  1, 78, 2376, 35640, 272160,  979776, 1306368,  279936;
  1, 84, 2808, 47520, 427680, 1959552, 3919104, 2239488;
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72.

Links

• Zagros Lalo, Left-justified triangle
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1+6x)^n

Crossrefs

Row sums give A015441.

Cf. A013613.

Programs

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 6 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten (* Robert G. Wilson v, May 19 2018 *)
Table[6^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

PARI

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1,k) + 6*T(n-2,k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018

Formula

T(n,k) = 6^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).