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A361432
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).
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2
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1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 20, 8, 0, 1, 5, 20, 54, 68, 16, 0, 1, 6, 30, 112, 252, 232, 32, 0, 1, 7, 42, 200, 656, 1188, 792, 64, 0, 1, 8, 56, 324, 1400, 3904, 5616, 2704, 128, 0, 1, 9, 72, 490, 2628, 10000, 23360, 26568, 9232, 256, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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T(0,k) = 1, T(1,k) = k; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n + (k - sqrt(k))^n)/2.
G.f. of column k: (1 - k * x)/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * cosh(sqrt(k) * x).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 6, 12, 20, 30, ...
0, 4, 20, 54, 112, 200, ...
0, 8, 68, 252, 656, 1400, ...
0, 16, 232, 1188, 3904, 10000, ...
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PROG
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(PARI) T(n, k) = sum(j=0, n\2, k^(n-j)*binomial(n, 2*j));
(PARI) T(n, k) = round(((k+sqrt(k))^n+(k-sqrt(k))^n))/2;
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CROSSREFS
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Column k=0..11 give A000007, A011782, A006012, A083881, A081335, A090139, A145301, A145302, A145303, A143079, A289414, A289415.
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KEYWORD
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AUTHOR
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STATUS
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approved
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