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