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%I #7 Sep 08 2022 08:45:38
%S 1,1,1,1,5,1,1,12,12,1,1,31,60,31,1,1,86,253,253,86,1,1,249,987,1478,
%T 987,249,1,1,736,3666,7325,7325,3666,736,1,1,2195,13150,32861,43810,
%U 32861,13150,2195,1,1,6570,45963,137865,229761,229761,137865,45963,6570,1
%N Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.
%C Row sums are: {1, 2, 7, 26, 124, 680, 3952, 23456, 140224, 840320, 5039872}.
%H G. C. Greubel, <a href="/A146987/b146987.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 12, 12, 1;
%e 1, 31, 60, 31, 1;
%e 1, 86, 253, 253, 86, 1;
%e 1, 249, 987, 1478, 987, 249, 1;
%p q:=3; seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + q^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # _G. C. Greubel_, Jan 09 2020
%t Table[If[n<2, Binomial[n, m], Binomial[n, m] + 3^(n-1)*Binomial[n-2, m-1]], {n, 0, 10}, {m, 0, n}]//Flatten
%o (PARI) T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + 3^(n-1)*binomial(n-2,k-1) ); \\ _G. C. Greubel_, Jan 09 2020
%o (Magma) T:= func< n,k,q | n lt 2 select Binomial(n,k) else Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1) >;
%o [T(n,k,3): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 09 2020
%o (Sage)
%o @CachedFunction
%o def T(n, k, q):
%o if (n<2): return binomial(n,k)
%o else: return binomial(n,k) + q^(n-1)*binomial(n-2,k-1)
%o [[T(n, k, 3) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 09 2020
%o (GAP)
%o T:= function(n,k,q)
%o if n<2 then return Binomial(n,k);
%o else return Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1);
%o fi; end; Flat(List([0..10], n-> List([0..n], k-> T(n,k,3) ))); # _G. C. Greubel_, Jan 09 2020
%Y Cf. A028262.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 04 2008
%E Edited by _G. C. Greubel_, Jan 09 2020
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