|
%I #40 Oct 19 2022 13:00:20
%S 1,2,2,4,12,4,8,52,52,8,16,196,416,196,16,32,684,2644,2644,684,32,64,
%T 2276,14680,26440,14680,2276,64,128,7340,74652,220280,220280,74652,
%U 7340,128,256,23172,357328,1623964,2643360,1623964,357328,23172,256,512,72076,1637860,10978444,27227908,27227908,10978444,1637860,72076,512
%N 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.
%C 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:
%C a\b 1.......2.......3.......4.......5.......6
%C -1 A144431
%C 0 A007318 A038208 A038221
%C 1 A008292 A256890 A257180 A257606 A257607
%C 2 A060187 A257609 A257611 A257613 A257615
%C 3 A142458 A257610 A257620 A257622 A257624 A257626
%C 4 A142459 A257612 A257621
%C 5 A142460 A257614 A257623
%C 6 A142461 A257616 A257625
%C 7 A142462 A257617 A257627
%C 8 A167884 A257618
%C 9 A257608 A257619
%C The row sums of these, and similarly constructed number triangles, are shown in the following table:
%C a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
%C 0 A000079 A000302 A000400
%C 1 A000142 A001715 A001725 A049388 A049198
%C 2 A000165 A002866 A002866 A051580 A051582
%C 3 A008544 A051578 A037559 A051605 A051607 A051609
%C 4 A001813 A047053 A000407 A034177 A051618 A051620 A051622
%C 5 A047055 A008546 A008548 A034300 A034325 A051688 A051690
%C 6 A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
%C 7 A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
%C 8 A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
%C 9 A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
%C 10 A051262 A035265 A035273 A035277
%C 11 A254322
%C 12 A145448
%C 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).
%C a\ b 0 1 2 3
%C -2 A130595/1
%C -1
%C 0
%C 1 A110555/-1 A120434/-1 A144697/1 A144699/2
%C 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.
%C If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
%C 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
%H Michael De Vlieger, <a href="/A256890/b256890.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)
%H L. Carlitz and R. Scoville, <a href="https://eudml.org/doc/151403">Generalized Eulerian numbers: combinatorial applications</a>, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
%H Dale Gerdemann, <a href="https://www.youtube.com/v=sKwtQYFTcY4">A256890, Plot of t(m,n) mod k </a>, YouTube, 2015.
%H Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, <a href="https://arxiv.org/abs/1807.01412">An asymptotic distribution theory for Eulerian recurrences with applications</a>, arXiv:1807.01412 [math.CO], 2018-2019.
%F 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.
%F Sum_{k=0..n} T(n, k) = A001715(n).
%F 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
%F 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
%F From _G. C. Greubel_, Oct 18 2022: (Start)
%F T(n, n-k) = T(n, k).
%F T(n, 0) = A000079(n). (End)
%e Array, t(n, k), begins as:
%e 1, 2, 4, 8, 16, 32, 64, ...;
%e 2, 12, 52, 196, 684, 2276, 7340, ...;
%e 4, 52, 416, 2644, 14680, 74652, 357328, ...;
%e 8, 196, 2644, 26440, 220280, 1623964, 10978444, ...;
%e 16, 684, 14680, 220280, 2643360, 27227908, 251195000, ...;
%e 32, 2276, 74652, 1623964, 27227908, 381190712, 4677894984, ...;
%e 64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
%e Triangle, T(n, k), begins as:
%e 1;
%e 2, 2;
%e 4, 12, 4;
%e 8, 52, 52, 8;
%e 16, 196, 416, 196, 16;
%e 32, 684, 2644, 2644, 684, 32;
%e 64, 2276, 14680, 26440, 14680, 2276, 64;
%e 128, 7340, 74652, 220280, 220280, 74652, 7340, 128;
%e 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256;
%t 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 *)
%o (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)));
%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ _Michel Marcus_, Apr 14 2015
%o (Magma)
%o A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
%o [A256890(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Oct 18 2022
%o (SageMath)
%o 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))
%o flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # _G. C. Greubel_, Oct 18 2022
%Y Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.
%Y Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.
%K nonn,tabl,easy
%O 0,2
%A _Dale Gerdemann_, Apr 12 2015
|
