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

%I A123490 #11 Sep 08 2022 08:45:28
%S A123490 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,
%T A123490 128,365,342,157,44,9,2,1,256,1094,1366,782,260,58,10,2,1,512,3281,
%U A123490 5462,3907,1556,401,74,11,2,1,1024,9842,21846,19532,9332,2802,586,92,12,2,1
%N A123490 Triangle whose k-th column satisfies a(n) = (k+3)*a(n-1)-(k+2)*a(n-2).
%H A123490 G. C. Greubel, <a href="/A123490/b123490.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A123490 Column k has g.f.: x^k*(1-x(1+k))/((1-x)*(1-x(2+k))).
%F A123490 T(n,k) = ((k+2)^(n-k) + k)/(k+1), for 0 <= k <= n.
%F A123490 Sum_{k=0..n} T(n, k) = A103439(n+1).
%F A123490 Sum_{k=0..floor(n/2)} T(n-k, k) = A123491(n).
%e A123490 Triangle begins
%e A123490      1;
%e A123490      2,    1;
%e A123490      4,    2,     1;
%e A123490      8,    5,     2,     1;
%e A123490     16,   14,     6,     2,    1;
%e A123490     32,   41,    22,     7,    2,    1;
%e A123490     64,  122,    86,    32,    8,    2,   1;
%e A123490    128,  365,   342,   157,   44,    9,   2,  1;
%e A123490    256, 1094,  1366,   782,  260,   58,  10,  2,  1;
%e A123490    512, 3281,  5462,  3907, 1556,  401,  74, 11,  2, 1;
%e A123490   1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1;
%t A123490 Table[((k+2)^(n-k) +k)/(k+1), {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 14 2017 *)
%o A123490 (PARI) for(n=0, 10, for(k=0,n, print1(((k+2)^(n-k)+k)/(k+1), ", "))) \\ _G. C. Greubel_, Oct 14 2017
%o A123490 (Magma) [((k+2)^(n-k) +k)/(k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 15 2021
%o A123490 (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
%Y A123490 Columns include A000079, A007051, A047849, A047850, A047851.
%Y A123490 Cf. A047848, A103439 (row sums), A123491 (diagonal sums).
%K A123490 easy,nonn,tabl
%O A123490 0,2
%A A123490 _Paul Barry_, Oct 01 2006