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A342134
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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x - k*x^2).
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4
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1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 18, 12, 0, 1, 8, 39, 80, 29, 0, 1, 10, 68, 252, 356, 70, 0, 1, 12, 105, 576, 1629, 1584, 169, 0, 1, 14, 150, 1100, 4880, 10530, 7048, 408, 0, 1, 16, 203, 1872, 11525, 41344, 68067, 31360, 985, 0, 1, 18, 264, 2940, 23364, 120750, 350272, 439992, 139536, 2378, 0
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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(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) + T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (1/2)^(n-j) * binomial(j,n-j).
T(n,k) = (-sqrt(k)*i)^n * U(n, sqrt(k)*i) where U(n, x) is a Chebyshev polynomial of the second kind.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
0, 5, 18, 39, 68, 105, ...
0, 12, 80, 252, 576, 1100, ...
0, 29, 356, 1629, 4880, 11525, ...
0, 70, 1584, 10530, 41344, 120750, ...
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MAPLE
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T:= (n, k)-> (<<0|1>, <k|2*k>>^(n+1))[1, 2]:
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MATHEMATICA
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T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * 2^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)
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PROG
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(PARI) T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*2^(-j)*binomial(n-j, j));
(PARI) T(n, k) = sum(j=0, n, (2*k)^j*2^(j-n)*binomial(j, n-j));
(PARI) T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I));
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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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