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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 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). 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 Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609. Cf. 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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