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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, 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
0,4
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-6x^2) are given by the sequence generated by the row sums.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72.
FORMULA
T(n,k) = 6^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).
EXAMPLE
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;
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, 12}, {k, 0, Floor[n/2]}] // Flatten (* Robert G. Wilson v, May 19 2018 *).
Table[6^k Binomial[n - k, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten.
PROG
(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
CROSSREFS
Row sums give A015441.
Cf. A013613.
Sequence in context: A342917 A040041 A162933 * A229085 A090850 A163945
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, May 09 2018
STATUS
approved