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A305695
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Triangle T(n,k) read by rows: fibonomial coefficients sums triangle.
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1
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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
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OFFSET
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0,2
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COMMENTS
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The triangle coefficients give sums of Fibonacci powers when multiplied with Lang triangle coefficients and summed (see 2nd formula).
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LINKS
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Table of n, a(n) for n=0..54.
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FORMULA
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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.
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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Cf. A000045, A010048, A000071, A056588, A317360.
Sequence in context: A131254 A210229 A210213 * A211235 A134626 A347633
Adjacent sequences: A305692 A305693 A305694 * A305696 A305697 A305698
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KEYWORD
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nonn,tabl
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AUTHOR
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Tony Foster III, Jul 09 2018
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EXTENSIONS
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More terms from Michel Marcus, Jul 20 2018
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STATUS
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approved
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