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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 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 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). 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, Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO] 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. 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 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 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 CROSSREFS Cf. A008292, A257180, A257606, A257607 Cf. A038208, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619 Sequence in context: A296688 A219569 A202795 * A110476 A330762 A059343 Adjacent sequences:  A256887 A256888 A256889 * A256891 A256892 A256893 KEYWORD nonn,tabl,easy AUTHOR Dale Gerdemann, Apr 12 2015 STATUS approved

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Last modified February 19 22:04 EST 2020. Contains 332060 sequences. (Running on oeis4.)