A317054 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 10 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 1, 1, 10, 1, 20, 1, 30, 100, 1, 40, 300, 1, 50, 600, 1000, 1, 60, 1000, 4000, 1, 70, 1500, 10000, 10000, 1, 80, 2100, 20000, 50000, 1, 90, 2800, 35000, 150000, 100000, 1, 100, 3600, 56000, 350000, 600000, 1, 110, 4500, 84000, 700000, 2100000, 1000000, 1, 120, 5500, 120000, 1260000, 5600000, 7000000

Offset

0,4

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013617 ((1+10x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038303 ((10+x)^n).

The coefficients in the expansion of 1/(1-x-10x^2) are given by the sequence generated by the row sums.

The row sums are Generalized Fibonacci numbers (see A015446).

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.701562118716424343244... ((1+sqrt(41))/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, 102.

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 + 10x)^n

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (10 + x)^n

Example

Triangle begins:
1;
1;
1, 10;
1, 20;
1, 30, 100;
1, 40, 300;
1, 50, 600, 1000;
1, 60, 1000, 4000;
1, 70, 1500, 10000, 10000;
1, 80, 2100, 20000, 50000;
1, 90, 2800, 35000, 150000, 100000;
1, 100, 3600, 56000, 350000, 600000;
1, 110, 4500, 84000, 700000, 2100000, 1000000;
1, 120, 5500, 120000, 1260000, 5600000, 7000000;
1, 130, 6600, 165000, 2100000, 12600000, 28000000, 10000000;
1, 140, 7800, 220000, 3300000, 25200000, 84000000, 80000000;
1, 150, 9100, 286000, 4950000, 46200000, 210000000, 360000000, 100000000;

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0,  t[n - 1, k] + 10 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten.
Table[10^k Binomial[n - k, k], {n, 0, 15}, {k, 0, Floor[n/2]}].

Prog

(PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k)+10*T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

Crossrefs

Row sums give A015446.

Cf. A013617

Cf. A038303

Sequence in context: A220448 A059920 A040109 * A292690 A362371 A362551

Adjacent sequences: A317051 A317052 A317053 * A317055 A317056 A317057

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 20 2018

Status

approved