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T(n,k) = T(n-1,k) + k*T(n-2,k) for k >= 1 and n >= 3 with T(0,k) = 0 and T(1,k) = T(2,k) = 1 for all k >= 1; array T(n,k), read by descending antidiagonals, with n >= 0 and k >= 1.
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%I #18 Dec 26 2019 12:55:16

%S 0,0,1,0,1,1,0,1,1,2,0,1,1,3,3,0,1,1,4,5,5,0,1,1,5,7,11,8,0,1,1,6,9,

%T 19,21,13,0,1,1,7,11,29,40,43,21,0,1,1,8,13,41,65,97,85,34,0,1,1,9,15,

%U 55,96,181,217,171,55,0,1,1,10,17,71,133,301,441,508,341,89,0,1,1,11

%N T(n,k) = T(n-1,k) + k*T(n-2,k) for k >= 1 and n >= 3 with T(0,k) = 0 and T(1,k) = T(2,k) = 1 for all k >= 1; array T(n,k), read by descending antidiagonals, with n >= 0 and k >= 1.

%C Transposed variant of A083856, without the top row of A083856.

%C Antidiagonal sums are (0, 1, 2, 4, 8, 16, 33, 70, 153, 345, ...) = (A110113(n) - 1: n >= 1).

%C Characteristic polynomials for columns are y^2 - y - k.

%e Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

%e 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ...

%e 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, ...

%e 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, ...

%e 13, 43, 97, 181, 301, 463, 673, 937, 1261, 1651, ...

%e 21, 85, 217, 441, 781, 1261, 1905, 2737, 3781, 5061, ...

%e 34, 171, 508, 1165, 2286, 4039, 6616, 10233, 15130, 21571, ...

%e 55, 341, 1159, 2929, 6191, 11605, 19951, 32129, 49159, 72181, ...

%e ...

%p A172237 := proc(n,k)

%p if n = 0 then

%p 0;

%p elif n <=2 then

%p 1 ;

%p else

%p procname(n-1,k)+k*procname(n-2,k) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jul 05 2012

%t f[0, a_] := 0; f[1, a_] := 1;

%t f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];

%t m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];

%t Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];

%t Flatten[%]

%Y Cf. A083856, A110113, A193376.

%K nonn,tabl,easy

%O 0,10

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jan 29 2010

%E More terms from _Petros Hadjicostas_, Dec 26 2019