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A317016
Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 7 * T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0.
1
1, 1, 1, 7, 1, 14, 1, 21, 49, 1, 28, 147, 1, 35, 294, 343, 1, 42, 490, 1372, 1, 49, 735, 3430, 2401, 1, 56, 1029, 6860, 12005, 1, 63, 1372, 12005, 36015, 16807, 1, 70, 1764, 19208, 84035, 100842, 1, 77, 2205, 28812, 168070, 352947, 117649, 1, 84, 2695, 41160, 302526, 941192, 823543
OFFSET
0,4
COMMENTS
The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013614 ((1+7*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A027466 ((7+x)^n).
The coefficients in the expansion of 1/(1-x-7x^2) are given by the sequence generated by the row sums.
The row sums are Generalized Fibonacci numbers (see A015442).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.192582403567252..., when n approaches infinity (see A223140).
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pages 70, 96.
FORMULA
T(n,k) = 7^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).
EXAMPLE
Triangle begins:
1;
1;
1, 7;
1, 14;
1, 21, 49;
1, 28, 147;
1, 35, 294, 343;
1, 42, 490, 1372;
1, 49, 735, 3430, 2401;
1, 56, 1029, 6860, 12005;
1, 63, 1372, 12005, 36015, 16807;
1, 70, 1764, 19208, 84035, 100842;
1, 77, 2205, 28812, 168070, 352947, 117649;
1, 84, 2695, 41160, 302526, 941192, 823543;
1, 91, 3234, 56595, 504210, 2117682, 3294172, 823543;
1, 98, 3822, 75460, 792330, 4235364, 9882516, 6588344;
MATHEMATICA
t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 7 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten.
Table[7^k Binomial[n - k, k], {n, 0, 15}, {k, 0, Floor[n/2]}].
PROG
(GAP) Flat(List([0..13], n->List([0..Int(n/2)], k->7^k*Binomial(n-k, k)))); # Muniru A Asiru, Jul 19 2018
CROSSREFS
Row sums give A015442.
Sequence in context: A225017 A268919 A040055 * A348983 A013614 A179530
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Jul 19 2018
STATUS
approved