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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.

%I A317052 #20 Aug 08 2025 07:05:06
%S A317052 1,9,81,1,729,18,6561,243,1,59049,2916,27,531441,32805,486,1,4782969,
%T A317052 354294,7290,36,43046721,3720087,98415,810,1,387420489,38263752,
%U A317052 1240029,14580,45,3486784401,387420489,14880348,229635,1215,1,31381059609,3874204890,172186884,3306744,25515,54
%N 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.
%C A317052 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).
%C A317052 The coefficients in the expansion of 1/(1-9*x-x^2) are given by the sequence generated by the row sums (see A099371).
%C A317052 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)).
%D A317052 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 100.
%H A317052 Zagros Lalo, <a href="/A317052/a317052.pdf">Left-justified triangle</a>
%H A317052 Zagros Lalo, <a href="/A317052/a317052_1.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 9x)^n</a>
%H A317052 Zagros Lalo, <a href="/A317052/a317052_2.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (9 + x)^n</a>
%e A317052 Triangle begins:
%e A317052   1;
%e A317052   9;
%e A317052   81, 1;
%e A317052   729, 18;
%e A317052   6561, 243, 1;
%e A317052   59049, 2916, 27;
%e A317052   531441, 32805, 486, 1;
%e A317052   4782969, 354294, 7290, 36;
%e A317052   43046721, 3720087, 98415, 810, 1;
%e A317052   387420489, 38263752, 1240029, 14580, 45;
%e A317052   3486784401, 387420489, 14880348, 229635, 1215, 1;
%e A317052   31381059609, 3874204890, 172186884, 3306744, 25515, 54;
%t A317052 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
%o A317052 (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)));
%o A317052 tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 20 2018
%Y A317052 Row sums give A099371.
%Y A317052 Cf. A013616, A038291, A176522.
%Y A317052 Cf. A001019 (column 0), A053540 (column 1), A081139 (column 2), A173187 (column 3), A173000 (column 4).
%K A317052 tabf,nonn,easy
%O A317052 0,2
%A A317052 _Zagros Lalo_, Jul 20 2018