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 A317052 Triangle read by rows: T(0,0) = 1; T(n,k) = 9 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, 9, 81, 1, 729, 18, 6561, 243, 1, 59049, 2916, 27, 531441, 32805, 486, 1, 4782969, 354294, 7290, 36, 43046721, 3720087, 98415, 810, 1, 387420489, 38263752, 1240029, 14580, 45, 3486784401, 387420489, 14880348, 229635, 1215, 1, 31381059609, 3874204890, 172186884, 3306744, 25515, 54 (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 A013616 ((1+9*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038291 ((9+x)^n). The coefficients in the expansion of 1/(1-9x-x^2) are given by the sequence generated by the row sums (see A099371). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 9.109772228646443655... (a metallic mean), when n approaches infinity; (see A176522: ((9+sqrt(85))/2)). 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 Zagros Lalo, Left-justified triangle EXAMPLE Triangle begins: 1; 9; 81, 1; 729, 18; 6561, 243, 1; 59049, 2916, 27; 531441, 32805, 486, 1; 4782969, 354294, 7290, 36; 43046721, 3720087, 98415, 810, 1; 387420489, 38263752, 1240029, 14580, 45; 3486784401, 387420489, 14880348, 229635, 1215, 1; 31381059609, 3874204890, 172186884, 3306744, 25515, 54; 282429536481, 38354628411, 1937102445, 44641044, 459270, 1701, 1; 2541865828329, 376572715308, 21308126895, 573956280, 7440174, 40824, 63; MATHEMATICA t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 9 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, 9*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 A099371. Cf. A013616 Cf. A038291 Cf. A176522 Cf. A001019 (column 0), A053540 (column 1), A081139 (column 2), A173187 (column 3), A173000 (column 4). Sequence in context: A127265 A055070 A143848 * A332702 A117817 A328760 Adjacent sequences:  A317049 A317050 A317051 * A317053 A317054 A317055 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Jul 20 2018 STATUS approved

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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)