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A365962
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Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).
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1
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1, 3, 1, 10, 6, 1, 48, 30, 9, 1, 312, 192, 60, 12, 1, 2520, 1560, 480, 100, 15, 1, 24480, 15120, 4680, 960, 150, 18, 1, 277200, 171360, 52920, 10920, 1680, 210, 21, 1, 3588480, 2217600, 685440, 141120, 21840, 2688, 280, 24, 1, 52254720, 32296320, 9979200, 2056320, 317520, 39312, 4032, 360, 27, 1
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f. of column k: (1+x)^2*x^k / ((1-x-x^2)*k!), k >= 0.
T(n,n) = 1 and T(n,k) = n!/k!*Fibonacci(n-k+3), n > k >= 0.
T(n,k) = n!/k!*Sum_{j=k..n} Fibonacci(j-k+1)*binomial(2,n-j).
T(n,k) = n!/k!*Sum_{j=k..n} (Fibonacci(j-k)+(-1)^(j-k))*binomial(3,n-j).
Recurrence: T(n,0) = A005921(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A341725(i,j)).
Sum_{j=k..n} (-1)^(n-j)*(n-j+1)!*binomial(n,j)*T(j,k) = A039948(n,k).
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EXAMPLE
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Triangle begins:
1,
3, 1,
10, 6, 1,
48, 30, 9, 1,
312, 192, 60, 12, 1,
2520, 1560, 480, 100, 15, 1,
24480, 15120, 4680, 960, 150, 18, 1,
...
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MAPLE
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T := proc(n, k) option remember; if k = n then 1 else (n!/k!*combinat[fibonacci](n-k+3)) fi end: seq(print(seq(T(n, k), k = 0..n)), n=0..9);
# second Maple program:
T := (n, k) -> add(Stirling2(j, k)*add(Stirling1(n, i)*A341725(i, j), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);
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PROG
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(PARI) T(n, k) = n!/k!*sum(j=k, n, fibonacci(j-k+1)*binomial(2, n-j)) \\ Winston de Greef, Oct 21 2023
(PARI) T(n, k) = if(n == k, 1, n!/k!*fibonacci(n-k+3)) \\ Winston de Greef, Oct 21 2023
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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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