A317051 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 9 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 1, 1, 9, 1, 18, 1, 27, 81, 1, 36, 243, 1, 45, 486, 729, 1, 54, 810, 2916, 1, 63, 1215, 7290, 6561, 1, 72, 1701, 14580, 32805, 1, 81, 2268, 25515, 98415, 59049, 1, 90, 2916, 40824, 229635, 354294, 1, 99, 3645, 61236, 459270, 1240029, 531441, 1, 108, 4455, 87480, 826686, 3306744, 3720087

Offset

0,4

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013616 ((1+9*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038291 ((9+x)^n).

The coefficients in the expansion of 1/(1-x-9x^2) are given by the sequence generated by the row sums.

The row sums are Generalized Fibonacci numbers (see A015445).

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.5413812651491... ((1+sqrt(37))/2), when n approaches infinity.

References

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 100

Links

Table of n, a(n) for n=0..55.

Zagros Lalo, Left-justified triangle

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 9x)^n

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (9 + x)^n

Formula

T(n,k) = 9^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

Example

Triangle begins:
1;
1;
1, 9;
1, 18;
1, 27, 81;
1, 36, 243;
1, 45, 486, 729;
1, 54, 810, 2916;
1, 63, 1215, 7290, 6561;
1, 72, 1701, 14580, 32805;
1, 81, 2268, 25515, 98415, 59049;
1, 90, 2916, 40824, 229635, 354294;
1, 99, 3645, 61236, 459270, 1240029, 531441;
1, 108, 4455, 87480, 826686, 3306744, 3720087;
1, 117, 5346, 120285, 1377810, 7440174, 14880348, 4782969;
1, 126, 6318, 160380, 2165130, 14880348, 44641044, 38263752;
1, 135, 7371, 208494, 3247695, 27280638, 111602610, 172186884, 43046721;

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0,  t[n - 1, k] + 9 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten.
Table[9^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->9^k*Binomial(n-k, k)))); # Muniru A Asiru, Jul 20 2018
(PARI) T(n, k) =  9^k*binomial(n-k, k);
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018
(Magma) /* As triangle */ [[9^k*Binomial(n-k, k): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018

Crossrefs

Row sums give A015445.

Cf. A013616

Cf. A038291

Sequence in context: A054018 A342637 A010170 * A013616 A205381 A237587

Adjacent sequences: A317048 A317049 A317050 * A317052 A317053 A317054

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 20 2018

Status

approved