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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 Zagros Lalo, Left-justified triangle 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. A013614, A027466, A176439. 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 Adjacent sequences:  A317011 A317012 A317013 * A317015 A317016 A317017 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Jul 19 2018 STATUS approved

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Last modified June 14 10:04 EDT 2021. Contains 345025 sequences. (Running on oeis4.)