|
%I #10 Sep 08 2022 08:45:38
%S 1,1,1,1,6,1,1,19,19,1,1,68,134,68,1,1,261,778,778,261,1,1,1030,4111,
%T 6164,4111,1030,1,1,4103,20501,40995,40995,20501,4103,1,1,16392,98332,
%U 245816,327750,245816,98332,16392,1,1,65545,458788,1376340,2293886,2293886,1376340,458788,65545,1
%N Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.
%C Row sums are {1, 2, 8, 40, 272, 2080, 16448, 131200, 1048832, 8389120, 67109888, ...} = {1, 2, 8*A081342(n)}. (modified by _G. C. Greubel_, Jan 09 2020)
%H G. C. Greubel, <a href="/A146988/b146988.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 4*(2^n + 8^n) otherwise. - _G. C. Greubel_, Jan 09 2020
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 19, 19, 1;
%e 1, 68, 134, 68, 1;
%e 1, 261, 778, 778, 261, 1;
%e 1, 1030, 4111, 6164, 4111, 1030, 1;
%p q:=4; 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] + 4^(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) + 4^(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,4): 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, 4) 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,4) ))); # _G. C. Greubel_, Jan 09 2020
%Y Cf. A028262, A081342.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 04 2008
%E Edited by _G. C. Greubel_, Jan 09 2020
|
