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A361290
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).
2
0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 4, 4, 0, 0, 1, 6, 14, 8, 0, 0, 1, 8, 30, 48, 16, 0, 0, 1, 10, 52, 144, 164, 32, 0, 0, 1, 12, 80, 320, 684, 560, 64, 0, 0, 1, 14, 114, 600, 1936, 3240, 1912, 128, 0, 0, 1, 16, 154, 1008, 4400, 11648, 15336, 6528, 256, 0
OFFSET
0,9
FORMULA
T(0,k) = 0, T(1,k) = 1; 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 * sqrt(k)) for k > 0.
G.f. of column k: x/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.
EXAMPLE
Square array begins:
0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1 , 1, ...
0, 2, 4, 6, 8, 10, ...
0, 4, 14, 30, 52, 80, ...
0, 8, 48, 144, 320, 600, ...
0, 16, 164, 684, 1936, 4400, ...
PROG
(PARI) T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);
CROSSREFS
Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.
Main diagonal gives A360766.
Cf. A361432.
Sequence in context: A072458 A256282 A258256 * A285638 A325667 A067310
KEYWORD
nonn,easy,tabl
AUTHOR
Seiichi Manyama, Mar 11 2023
STATUS
approved