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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2. 24
1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums are 1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800, 79833600, ... (see A001715)

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:

a\b 1.......2.......3.......4.......5.......6

-1  A144431

0   A007318 A038208 A038221

1   A008292 A256890 A257180 A257606 A257607

2   A060187 A257609 A257611 A257613 A257615

3   A142458 A257610 A257620 A257622 A257624 A257626

4   A142459 A257612 A257621

5   A142460 A257614 A257623

6   A142461 A257616 A257625

7   A142462 A257617 A257627

8   A167884 A257618

9   A257608 A257619

The row sums of these, and similarly constructed number triangles, are shown in the following table:

a\b 1.......2.......3.......4.......5.......6.......7.......8.......9

0   A000079 A000302 A000400

1   A000142 A001715 A001725 A049388 A049198

2   A000165 A002866 A002866 A051580 A051582

3   A008544 A051578 A037559 A051605 A051607 A051609

4   A001813 A047053 A000407 A034177 A051618 A051620 A051622

5   A047055 A008546 A008548 A034300 A034325 A051688 A051690

6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103

7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105

8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115

9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023

10                                  A051262 A035265 A035273         A035277

11                                  A254322

12                                          A145448

The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).

a\ b  0           1           2          3

-2    A130595/1

-1

0

1     A110555/-1  A120434/-1  A144697/1  A144699/2

With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4,, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash.

If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).

LINKS

Table of n, a(n) for n=0..54.

Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

EXAMPLE

1

2 2

4 12 4

8 52 52 8

16 196 416 196 16

32 684 2644 2644 684 32

64 2276 14680 26440 14680 2276 64

128 7340 74652 220280 220280 74652 7340 128

256 23172 357328 1623964 2643360 1623964 357328 23172 256

512 72076 1637860 10978444 27227908 27227908 10978444 1637860 72076 512

PROG

(PARI) t(n, m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", "); ); print(); ); } \\ Michel Marcus, Apr 14 2015

CROSSREFS

Cf. A008292, A257180, A257606, A257607

Cf. A038208, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619

Sequence in context: A296688 A219569 A202795 * A110476 A059343 A285944

Adjacent sequences:  A256887 A256888 A256889 * A256891 A256892 A256893

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, Apr 12 2015

STATUS

approved

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Last modified November 16 17:06 EST 2018. Contains 317274 sequences. (Running on oeis4.)