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A371026
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Triangle read by rows: T(n, k) = 4^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/4, n).
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3
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1, 0, 1, 0, 5, 2, 0, 45, 30, 6, 0, 585, 510, 180, 24, 0, 9945, 10350, 4950, 1200, 120, 0, 208845, 247590, 144900, 48600, 9000, 720, 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040, 0, 151412625, 216093150, 164872260, 80713080, 26334000, 5594400, 705600, 40320
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = n! for n >= 0. - Werner Schulte, Mar 17 2024
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EXAMPLE
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Triangle read by rows:
[0] 1;
[1] 0, 1;
[2] 0, 5, 2;
[3] 0, 45, 30, 6;
[4] 0, 585, 510, 180, 24;
[5] 0, 9945, 10350, 4950, 1200, 120;
[6] 0, 208845, 247590, 144900, 48600, 9000, 720;
[7] 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040;
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MAPLE
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A371026 := (n, k) -> local j; 4^n*add((-1)^(k - j)*binomial(k, j)*pochhammer(j/4, n), j = 0..k): seq(seq(A371026(n, k), k = 0..n), n = 0..9);
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PROG
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(Python)
from functools import cache
@cache
if k == 0: return 0**n
if k == n: return n * T(n-1, n-1)
return k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k)
for n in range(8): print([T(n, k) for k in range(n + 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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