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 A305695 Triangle T(n,k) read by rows: fibonomial coefficients sums triangle. 1
 1, 2, 1, 4, 3, 1, 7, 9, 4, 1, 12, 24, 19, 6, 1, 20, 64, 79, 46, 9, 1, 33, 168, 339, 306, 113, 14, 1, 54, 441, 1431, 2126, 1205, 287, 22, 1, 88, 1155, 6072, 14502, 13581, 4928, 736, 35, 1, 143, 3025, 25707, 99587, 149717, 90013, 20371, 1905, 56, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The triangle coefficients give sums of Fibonacci powers when multiplied with Lang triangle coefficients and summed (see 2nd formula). LINKS Table of n, a(n) for n=0..54. FORMULA T(n, k) = T(n-1, k) + A010048(n+1, k+1). Sum_{t=0..n-1} A056588(n-1, n-1-t) * T(k+t, n-1) = Sum_{j=1..k+1} F(j)^n. EXAMPLE n\k| 0 1 2 3 4 5 6 7 8 9 ---+-------------------------------------------------- 0 | 1 1 | 2 1 2 | 4 3 1 3 | 7 9 4 1 4 | 12 24 19 6 1 5 | 20 64 79 46 9 1 6 | 33 168 339 306 113 14 1 7 | 54 441 1431 2126 1205 287 22 1 8 | 88 1155 6072 14502 13581 4928 736 35 1 9 | 143 3025 25707 99587 149717 90013 20371 1905 56 1 PROG (PARI) f(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)); T(n, k) = if (n< 0, 0, T(n-1, k) + f(n+1, k+1)); tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018 CROSSREFS Cf. A000045, A010048, A000071, A056588, A317360. Sequence in context: A131254 A210229 A210213 * A211235 A134626 A347633 Adjacent sequences: A305692 A305693 A305694 * A305696 A305697 A305698 KEYWORD nonn,tabl AUTHOR Tony Foster III, Jul 09 2018 EXTENSIONS More terms from Michel Marcus, Jul 20 2018 STATUS approved

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Last modified September 11 02:28 EDT 2024. Contains 375813 sequences. (Running on oeis4.)