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 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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 Zagros Lalo, Left-justified triangle 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: A014721 A054018 A010170 * A013616 A205381 A237587 Adjacent sequences:  A317048 A317049 A317050 * A317052 A317053 A317054 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Jul 20 2018 STATUS approved

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Last modified June 6 13:31 EDT 2020. Contains 334827 sequences. (Running on oeis4.)