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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
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. A000420 (column 0), A027473 (column 1), A027474 (column 2), A140107 (column 3), A139641 (column 4).
Sequence in context: A024092 A163713 A192897 * A330329 A115589 A014390
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Jul 19 2018
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)