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%I #15 Sep 05 2018 02:27:35
%S 1,7,49,1,343,14,2401,147,1,16807,1372,21,117649,12005,294,1,823543,
%T 100842,3430,28,5764801,823543,36015,490,1,40353607,6588344,352947,
%U 6860,35,282475249,51883209,3294172,84035,735,1,1977326743,403536070,29647548,941192,12005,42
%N 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.
%C 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).
%C The coefficients in the expansion of 1/(1-7x-x^2) are given by the sequence generated by the row sums.
%C 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.
%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 96.
%H Zagros Lalo, <a href="/A317014/a317014.pdf">Left-justified triangle</a>
%H Zagros Lalo, <a href="/A317014/a317014_1.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n</a>
%H Zagros Lalo, <a href="/A317014/a317014_2.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n</a>
%e Triangle begins:
%e 1;
%e 7;
%e 49, 1;
%e 343, 14;
%e 2401, 147, 1;
%e 16807, 1372, 21;
%e 117649, 12005, 294, 1;
%e 823543, 100842, 3430, 28;
%e 5764801, 823543, 36015, 490, 1;
%e 40353607, 6588344, 352947, 6860, 35;
%e 282475249, 51883209, 3294172, 84035, 735, 1;
%e 1977326743, 403536070, 29647548, 941192, 12005, 42;
%e 13841287201, 3107227739, 259416045, 9882516, 168070, 1029, 1;
%e 96889010407, 23727920916, 2219448385, 98825160, 2117682, 19208, 49;
%e 678223072849, 179936733613, 18643366434, 951192165, 24706290, 302526, 1372, 1;
%t 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
%o (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)));
%o tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 20 2018
%Y Row sums give A054413.
%Y Cf. A013614, A027466, A176439.
%Y Cf. A000420 (column 0), A027473 (column 1), A027474 (column 2), A140107 (column 3), A139641 (column 4).
%K tabf,nonn,easy
%O 0,2
%A _Zagros Lalo_, Jul 19 2018
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