A172237 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.

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

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Jan 29 2010

Keywords

• nonn
• tabl
• easy

Comments

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

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

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

Examples

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
    0,    0,    0,    0,    0,     0,     0,     0,     0,     0, ...
    1,    1,    1,    1,    1,     1,     1,     1,     1,     1, ...
    1,    1,    1,    1,    1,     1,     1,     1,     1,     1, ...
    2,    3,    4,    5,    6,     7,     8,     9,    10,    11, ...
    3,    5,    7,    9,   11,    13,    15,    17,    19,    21, ...
    5,   11,   19,   29,   41,    55,    71,    89,   109,   131, ...
    8,   21,   40,   65,   96,   133,   176,   225,   280,   341, ...
   13,   43,   97,  181,  301,   463,   673,   937,  1261,  1651, ...
   21,   85,  217,  441,  781,  1261,  1905,  2737,  3781,  5061, ...
   34,  171,  508, 1165, 2286,  4039,  6616, 10233, 15130, 21571, ...
   55,  341, 1159, 2929, 6191, 11605, 19951, 32129, 49159, 72181, ...
   ...
		

Crossrefs

Cf. A083856, A110113, A193376.

Programs

Maple

A172237 := proc(n,k)
        if n = 0 then
                0;
        elif n <=2 then
                1 ;
        else
                procname(n-1,k)+k*procname(n-2,k) ;
        end if;
end proc: # R. J. Mathar, Jul 05 2012

Mathematica

f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

Extensions

More terms from Petros Hadjicostas, Dec 26 2019