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A144441
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Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
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8
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1, 1, 1, 1, 14, 1, 1, 83, 83, 1, 1, 432, 1550, 432, 1, 1, 2181, 19898, 19898, 2181, 1, 1, 10930, 217887, 523548, 217887, 10930, 1, 1, 54679, 2199237, 10589795, 10589795, 2199237, 54679, 1, 1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1
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OFFSET
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1,5
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LINKS
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FORMULA
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T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = 2*(2*n-3)*s(n-1) + 4*s(n-2) with s(1) = 1, s(2) = 2.
T(n, n-k) = T(n, k).
T(n, 2) = (1/2)*(7*5^(n-2) - (2*n+1)).
T(n, 3) = (1/8)*(4*n^2 - 5 - 14*(10*n-3)*5^(n-3) + 355*9^(n-3)). (End)
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 14, 1;
1, 83, 83, 1;
1, 432, 1550, 432, 1;
1, 2181, 19898, 19898, 2181, 1;
1, 10930, 217887, 523548, 217887, 10930, 1;
1, 54679, 2199237, 10589795, 10589795, 2199237, 54679, 1;
1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1;
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MATHEMATICA
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T[n_, k_, m_, j_]:= T[n, k, m, j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m, j] + (m*(k-1)+1)*T[n-1, k, m, j] + j*T[n-2, k-1, m, j]];
Table[T[n, k, 4, 4], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)
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PROG
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(Sage)
def T(n, k, m, j):
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*T(n-1, k-1, m, j) + (m*(k-1)+1)*T(n-1, k, m, j) + j*T(n-2, k-1, m, j)
def A144441(n, k): return T(n, k, 4, 4)
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CROSSREFS
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Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144442, A144443, A144444, A144445.
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KEYWORD
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AUTHOR
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STATUS
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approved
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