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A256890
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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.
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24
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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
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OFFSET
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0,2
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COMMENTS
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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
-1
0
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
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LINKS
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FORMULA
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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
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
T(n, n-k) = T(n, k).
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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]]) >;
(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))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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