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%I #15 Sep 08 2022 08:45:28
%S 1,2,1,6,4,1,18,16,6,1,54,60,30,8,1,162,216,134,48,10,1,486,756,558,
%T 248,70,12,1,1458,2592,2214,1168,410,96,14,1,4374,8748,8478,5160,2150,
%U 628,126,16,1,13122,29160,31590,21744,10442,3624,910,160,18,1
%N Riordan array ((1-x)/(1-3*x), x*(1-x)/(1-3*x)).
%C Row sums are A001835(n+1). Diagonal sums are A030186. Inverse is A125694. Equal to product of A007318 and A073370.
%H G. C. Greubel, <a href="/A125693/b125693.txt">Rows n=0..100 of triangle, flattened</a>
%F Number triangle T(n,k) = Sum_{j=0..k+1} C(k+1,j)*C(n-j,n-k-j)* (-1)^j * 3^(n-k-j).
%F T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k>n or if k<n. - _Philippe Deléham_, Jan 08 2013
%e Triangle begins
%e 1;
%e 2, 1;
%e 6, 4, 1;
%e 18, 16, 6, 1;
%e 54, 60, 30, 8, 1;
%e 162, 216, 134, 48, 10, 1;
%p seq(seq( add( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j), j=0..n), k=0..n), n=0..10); # _G. C. Greubel_, Oct 28 2019
%t T[0, 0]=1; T[1, 0]=2; T[1, 1]=1; T[n_, k_]/; 0<=k<=n:= T[n, k]= 3T[n-1, k] + T[n-1, k-1] - T[n-2, k-1]; T[_, _]=0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%t T[n_, k_]:= Sum[(-1)^j*3^(n-k-j)*Binomial[k+1,j]*Binomial[n-j,n-k-j], {j, 0, n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 28 2019 *)
%o (PARI) T(n,k) = sum(j=0,n, (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j));
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 28 2019
%o (Magma) T:= func< n,k | &+[(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j): j in [0..n]] >;
%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Oct 28 2019
%o (Sage) [[sum( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j) for j in (0..n) ) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Oct 28 2019
%o (GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j->
%o (-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j) )))); # _G. C. Greubel_, Oct 28 2019
%K nonn,tabl,easy
%O 0,2
%A _Paul Barry_, Nov 30 2006
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