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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 A115450

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 February 18 05:30 EST 2019. Contains 320245 sequences. (Running on oeis4.)