The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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
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
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
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).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.
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.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)
EXAMPLE
Array, t(n, k), begins as:
1, 2, 4, 8, 16, 32, 64, ...;
2, 12, 52, 196, 684, 2276, 7340, ...;
4, 52, 416, 2644, 14680, 74652, 357328, ...;
8, 196, 2644, 26440, 220280, 1623964, 10978444, ...;
16, 684, 14680, 220280, 2643360, 27227908, 251195000, ...;
32, 2276, 74652, 1623964, 27227908, 381190712, 4677894984, ...;
64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
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;
MATHEMATICA
Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j, 0, k}], {n, 0, 9}, {k, 0, n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)
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
(Magma)
A256890:= func< n, k | (&+[(-1)^(k-j)*Binomial(j+3, j)*Binomial(n+4, k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022
(SageMath)
def A256890(n, k): return sum((-1)^(k-j)*Binomial(j+3, j)*Binomial(n+4, k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n, k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022
CROSSREFS
Sequence in context: A296688 A219569 A202795 * A110476 A330762 A059343
KEYWORD
nonn,tabl,easy
AUTHOR
Dale Gerdemann, Apr 12 2015
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 28 00:20 EDT 2024. Contains 372900 sequences. (Running on oeis4.)