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%I #11 Sep 08 2022 08:45:38
%S 1,1,1,1,4,1,1,7,7,1,1,12,22,12,1,1,21,58,58,21,1,1,38,143,212,143,38,
%T 1,1,71,341,675,675,341,71,1,1,136,796,1976,2630,1976,796,136,1,1,265,
%U 1828,5460,9086,9086,5460,1828,265,1,1,522,4141,14456,28882,36092,28882,14456,4141,522,1
%N Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
%C Row sums are: {1, 2, 6, 16, 48, 160, 576, 2176, 8448, 33280, 132096, ...} = {1, 2, 2*A242985(n)}. (modified by _G. C. Greubel_, Jan 09 2020)
%H G. C. Greubel, <a href="/A146986/b146986.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
%F Sum_{k=0..n} T(n,k) = n+1 for n < 2 and 16*binomial(2^(n-3) + 1, 2) otherwise. - _G. C. Greubel_, Jan 09 2020
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 7, 7, 1;
%e 1, 12, 22, 12, 1;
%e 1, 21, 58, 58, 21, 1;
%e 1, 38, 143, 212, 143, 38, 1;
%p q:=2; 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] + 2^(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) + 2^(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,2): 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, 2) 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;
%o Flat(List([0..10], n-> List([0..n], k-> T(n,k,2) ))); # _G. C. Greubel_, Jan 09 2020
%Y Cf. A028262, A242985.
%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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