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
-1 A144431
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
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
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.
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
KEYWORD
AUTHOR
Dale Gerdemann, Apr 12 2015
STATUS
approved