A253273 - OEIS

%I #19 Apr 17 2021 21:48:27

%S 1,1,2,1,3,3,1,4,7,4,1,5,12,14,5,1,6,18,30,25,6,1,7,25,53,66,41,7,1,8,

%T 33,84,136,132,63,8,1,9,42,124,244,315,245,92,9,1,10,52,174,400,636,

%U 673,428,129,10,1,11,63,235,615,1152,1522,1346,711,175,11

%N Triangle T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1), read by rows.

%H G. C. Greubel, <a href="/A253273/b253273.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1).

%F Sum_{k=0..n} T(n,k) = A095263(n+1).

%F G.f.: 1/( (1-x)*(1+y^2) - (2-x)*y ).

%e The triangle begins as:

%e   1;

%e   1,  2;

%e   1,  3,  3;

%e   1,  4,  7,   4;

%e   1,  5, 12,  14,   5;

%e   1,  6, 18,  30,  25,   6;

%e   1,  7, 25,  53,  66,  41,   7;

%e   1,  8, 33,  84, 136, 132,  63,   8;

%e   1,  9, 42, 124, 244, 315, 245,  92,   9;

%e   1, 10, 52, 174, 400, 636, 673, 428, 129, 10;

%e   ...

%t T[n_, k_]:= Sum[Binomial[k+j,k-j+1]*Binomial[n-k,j-1], {j,0,n-k+1}];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 17 2021 *)

%o (Maxima)

%o T(n,m):=sum(binomial(m+k,m-k+1)*binomial(n-m,k-1),k,0,n-m+1);

%o (Magma)

%o T:= func< n,k | (&+[Binomial(k+j,k-j+1)*Binomial(n-k,j-1): j in [0..n-k+1]]) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 17 2021

%o (Sage)

%o def T(n,k): return sum(binomial(k+j,k-j+1)*binomial(n-k,j-1) for j in (0..n-k+1))

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 17 2021

%Y Cf. A095263.

%K nonn,tabl

%O 0,3

%A _Vladimir Kruchinin_, May 01 2015