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A317014 Triangle read by rows: T(0,0) = 1; T(n,k) = 7 * T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0. 2
1, 7, 49, 1, 343, 14, 2401, 147, 1, 16807, 1372, 21, 117649, 12005, 294, 1, 823543, 100842, 3430, 28, 5764801, 823543, 36015, 490, 1, 40353607, 6588344, 352947, 6860, 35, 282475249, 51883209, 3294172, 84035, 735, 1, 1977326743, 403536070, 29647548, 941192, 12005, 42 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013614 ((1+7*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A027466 ((7+x)^n).

The coefficients in the expansion of 1/(1-7x-x^2) are given by the sequence generated by the row sums.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 7.14005494464025913554... ((7+sqrt(53))/2), a metallic mean (see A176439), 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, 96.

LINKS

Table of n, a(n) for n=0..41.

Zagros Lalo, Left-justified triangle

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n

EXAMPLE

Triangle begins:

1;

7;

49, 1;

343, 14;

2401, 147, 1;

16807, 1372, 21;

117649, 12005, 294, 1;

823543, 100842, 3430, 28;

5764801, 823543, 36015, 490, 1;

40353607, 6588344, 352947, 6860, 35;

282475249, 51883209, 3294172, 84035, 735, 1;

1977326743, 403536070, 29647548, 941192, 12005, 42;

13841287201, 3107227739, 259416045, 9882516, 168070, 1029, 1;

96889010407, 23727920916, 2219448385, 98825160, 2117682, 19208, 49;

678223072849, 179936733613, 18643366434, 951192165, 24706290, 302526, 1372, 1;

MATHEMATICA

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 7 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

PROG

(PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 7*T(n-1, k)+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 A054413.

Cf. A013614, A027466, A176439.

Cf. A000420 (column 0), A027473 (column 1), A027474 (column 2), A140107 (column 3), A139641 (column 4).

Sequence in context: A024092 A163713 A192897 * A330329 A115589 A014390

Adjacent sequences:  A317011 A317012 A317013 * A317015 A317016 A317017

KEYWORD

tabf,nonn,easy

AUTHOR

Zagros Lalo, Jul 19 2018

STATUS

approved

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Last modified June 14 10:04 EDT 2021. Contains 345025 sequences. (Running on oeis4.)