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Triangular array T given by T(n,k) = Sum_{0 <= j <= i-n+k, n-k <= i <= n} C(i,j) for n >= 0 and 0 <= k <= n.
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%I #45 Sep 08 2022 08:45:00

%S 1,1,3,1,4,7,1,5,11,15,1,6,16,26,31,1,7,22,42,57,63,1,8,29,64,99,120,

%T 127,1,9,37,93,163,219,247,255,1,10,46,130,256,382,466,502,511,1,11,

%U 56,176,386,638,848,968,1013,1023,1,12,67,232,562,1024,1486,1816,1981,2036,2047

%N Triangular array T given by T(n,k) = Sum_{0 <= j <= i-n+k, n-k <= i <= n} C(i,j) for n >= 0 and 0 <= k <= n.

%C Row sums given by A001787.

%C T(n, n) = -1 + 2^(n+1).

%C T(2*n, n) = 4^n.

%C T(2*n+1, n) = A000346(n).

%C T(2*n-1, n) = A032443(n).

%C A054143 is the fission of the polynomial sequence ((x+1^n) by the polynomial sequence (q(n,x)) given by q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - _Clark Kimberling_, Aug 07 2011

%H G. C. Greubel, <a href="/A054143/b054143.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = Sum_{0 <= j <= i-n+k, n-k <= i <= n} binomial(i,j).

%F T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 30 2013

%F From _Petros Hadjicostas_, Jun 05 2020: (Start)

%F Bivariate o.g.f.: Sum_{n,k >= 0} T(n,k)*x^n*y^k = 1/(1 - x - 3*x*y + 2*x^2*y + 2*x^2*y^2) = 1/((1 - 2*x*y)*(1 - x*(y+1))).

%F n-th row o.g.f.: ((1 + y)^(n+1) - (2*y)^(n+1))/(1 - y). (End)

%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

%e 1;

%e 1, 3;

%e 1, 4, 7;

%e 1, 5, 11, 15;

%e 1, 6, 16, 26, 31;

%e 1, 7, 22, 42, 57, 63;

%p A054143_row := proc(n) add(add(binomial(n,n-i)*x^(k+1),i=0..k),k=0..n-1); coeffs(sort(%)) end; seq(print(A054143_row(n)),n=1..6); # _Peter Luschny_, Sep 29 2011

%t (* First program *)

%t z=10;

%t p[n_,x_]:=(x+1)^n;

%t q[0,x_]:=1;q[n_,x_]:=x*q[n-1,x]+1;

%t p1[n_,k_]:=Coefficient[p[n,x],x^k];p1[n_,0]:=p[n,x]/.x->0;

%t d[n_,x_]:=Sum[p1[n,k]*q[n-1-k,x],{k,0,n-1}]

%t h[n_]:=CoefficientList[d[n,x],{x}]

%t TableForm[Table[Reverse[h[n]],{n,0,z}]]

%t Flatten[Table[Reverse[h[n]],{n,-1,z}]] (* A054143 *)

%t TableForm[Table[h[n],{n,0,z}]]

%t Flatten[Table[h[n],{n,-1,z}]] (* A104709 *)

%t (* Second program *)

%t Table[Sum[Binomial[i, j], {i, n-k, n}, {j,0,i-n+k}], {n,0,12}, {k,0,n}]// Flatten (* _G. C. Greubel_, Aug 01 2019 *)

%o (PARI) T(n,k) = sum(i=n-k,n, sum(j=0,i-n+k, binomial(i,j)));

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Aug 01 2019

%o (Magma)

%o T:= func< n,k | (&+[ (&+[ Binomial(i,j): j in [0..i-n+k]]): i in [n-k..n]]) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 01 2019

%o (Sage)

%o def T(n, k): return sum(sum( binomial(i,j) for j in (0..i-n+k)) for i in (n-k..n))

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 01 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([n-k..n], i-> Sum([0..i-n+k], j-> Binomial(i,j) ))))); # _G. C. Greubel_, Aug 01 2019

%Y Cf. A000346, A001787, A032443.

%Y Diagonal sums give A005672. - _Paul Barry_, Feb 07 2003

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Mar 18 2000

%E Name edited by _Petros Hadjicostas_, Jun 04 2020