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A360441
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Triangle read by rows: T(n,k) is the number of pairs (c,m), where c is a covering of the 1 X (2n) grid with 1 X 2 rectangles and equal numbers of red and blue 1 X 1 squares and m is a matching between red squares and blue squares, such that exactly k matched pairs are adjacent.
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0
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1, 1, 2, 7, 8, 4, 71, 78, 36, 8, 1001, 1072, 504, 128, 16, 18089, 19090, 9080, 2480, 400, 32, 398959, 417048, 199980, 56960, 10320, 1152, 64, 10391023, 10789982, 5204556, 1523480, 295120, 38304, 3136, 128, 312129649, 322520672, 156264304, 46629632, 9436000, 1336832, 130816, 8192, 256
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OFFSET
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0,3
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COMMENTS
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If row elements are divided by row sums, one obtains a probability distribution that approaches a Poisson distribution with expected value 1 as n approaches infinity.
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LINKS
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FORMULA
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T(n,k) equals 2^k times the corresponding element of the triangle of A168422.
T(n,k) = 2^k * Sum_{j=k..n} (-1)^(j-k) * C(2*n-j,n) * C(n,j) * C(j,k) * (n-j)!.
Recurrence: T(n,k) = (1/k!) * Sum_{j=0..k} T(n-j,0) * (-1)^j * C(k,j) * Sum_{t=0..min(j,k-j)} (-1)^(j-t) * C(j,t) * (k-j)! / (k-j-t)!
= (1/k!) * Sum_{j=0..k} T(n-j,0) * (-1)^j * C(k,j) * R(k,j) where R(k,j) is an element of the triangle of A253667.
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EXAMPLE
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Triangle begins:
1
1 2
7 8 4
71 78 36 8
1001 1072 504 128 16
18089 19090 9080 2480 400 32
398959 417048 199980 56960 10320 1152 64
10391023 10789982 5204556 1523480 295120 38304 3136 128
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PROG
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(SageMath)
def T(n, k):
return(2^k*sum((-1)^(j-k)*binomial(2*n-j, n)*binomial(n, j)\
*binomial(j, k)*factorial(n-j) for j in range(k, n+1)))
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CROSSREFS
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T(n,k) equals 2^k times the corresponding element of the triangle of A168422.
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KEYWORD
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AUTHOR
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STATUS
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approved
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