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 A317054 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 10 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. 2
 1, 1, 1, 10, 1, 20, 1, 30, 100, 1, 40, 300, 1, 50, 600, 1000, 1, 60, 1000, 4000, 1, 70, 1500, 10000, 10000, 1, 80, 2100, 20000, 50000, 1, 90, 2800, 35000, 150000, 100000, 1, 100, 3600, 56000, 350000, 600000, 1, 110, 4500, 84000, 700000, 2100000, 1000000, 1, 120, 5500, 120000, 1260000, 5600000, 7000000 (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 A013617 ((1+10x)^n) and  along skew diagonals pointing top-left in center-justified triangle given in A038303 ((10+x)^n). The coefficients in the expansion of 1/(1-x-10x^2) are given by the sequence generated by the row sums. The row sums are Generalized Fibonacci numbers (see A015446). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.701562118716424343244... ((1+sqrt(41))/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, 102. LINKS Zagros Lalo, Left-justified triangle MAPLE Triangle begins: 1; 1; 1, 10; 1, 20; 1, 30, 100; 1, 40, 300; 1, 50, 600, 1000; 1, 60, 1000, 4000; 1, 70, 1500, 10000, 10000; 1, 80, 2100, 20000, 50000; 1, 90, 2800, 35000, 150000, 100000; 1, 100, 3600, 56000, 350000, 600000; 1, 110, 4500, 84000, 700000, 2100000, 1000000; 1, 120, 5500, 120000, 1260000, 5600000, 7000000; 1, 130, 6600, 165000, 2100000, 12600000, 28000000, 10000000; 1, 140, 7800, 220000, 3300000, 25200000, 84000000, 80000000; 1, 150, 9100, 286000, 4950000, 46200000, 210000000, 360000000, 100000000; MATHEMATICA t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0,  t[n - 1, k] + 10 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten. Table[10^k Binomial[n - k, k], {n, 0, 15}, {k, 0, Floor[n/2]}]. PROG (PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k)+10*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 A015446. Cf. A013617 Cf. A038303 Sequence in context: A220448 A059920 A040109 * A292690 A036188 A013617 Adjacent sequences:  A317051 A317052 A317053 * A317055 A317056 A317057 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Jul 20 2018 STATUS approved

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Last modified May 15 18:26 EDT 2021. Contains 343920 sequences. (Running on oeis4.)