A123490 Triangle whose k-th column satisfies a(n) = (k+3)*a(n-1)-(k+2)*a(n-2).

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 16, 14, 6, 2, 1, 32, 41, 22, 7, 2, 1, 64, 122, 86, 32, 8, 2, 1, 128, 365, 342, 157, 44, 9, 2, 1, 256, 1094, 1366, 782, 260, 58, 10, 2, 1, 512, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1

Offset: 0

Author

Paul Barry, Oct 01 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
     1;
     2,    1;
     4,    2,     1;
     8,    5,     2,     1;
    16,   14,     6,     2,    1;
    32,   41,    22,     7,    2,    1;
    64,  122,    86,    32,    8,    2,   1;
   128,  365,   342,   157,   44,    9,   2,  1;
   256, 1094,  1366,   782,  260,   58,  10,  2,  1;
   512, 3281,  5462,  3907, 1556,  401,  74, 11,  2, 1;
  1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Columns include A000079, A007051, A047849, A047850, A047851.

Cf. A047848, A103439 (row sums), A123491 (diagonal sums).

Programs

Magma

[((k+2)^(n-k) +k)/(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 15 2021

Mathematica

Table[((k+2)^(n-k) +k)/(k+1), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 14 2017 *)

PARI

for(n=0, 10, for(k=0,n, print1(((k+2)^(n-k)+k)/(k+1), ", "))) \\ G. C. Greubel, Oct 14 2017

Sage

flatten([[((k+2)^(n-k) +k)/(k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Formula

Column k has g.f.: x^k*(1-x(1+k))/((1-x)*(1-x(2+k))).

T(n,k) = ((k+2)^(n-k) + k)/(k+1), for 0 <= k <= n.

Sum_{k=0..n} T(n, k) = A103439(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A123491(n).