A171824 - OEIS

%I #10 Apr 30 2021 02:23:10

%S 2,3,3,7,6,7,21,14,14,21,71,40,30,40,71,253,132,77,77,132,253,925,469,

%T 238,168,238,469,925,3433,1724,828,450,450,828,1724,3433,12871,6444,

%U 3048,1452,990,1452,3048,6444,12871,48621,24320,11495,5225,2717,2717,5225,11495,24320,48621

%N Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.

%H G. C. Greubel, <a href="/A171824/b171824.txt">Table of n, a(n) for n = 0..1325</a>

%F T(n,k) = A046899(n,k) + A092392(n,k).

%F Sum_{k=0..n} T(n,k) = binomial(2*n+2, n+1) = 2*A001700(n) = A000984(n+1). - _G. C. Greubel_, Apr 29 2021

%e Triangle begins as:

%e        2;

%e        3,     3;

%e        7,     6,     7;

%e       21,    14,    14,    21;

%e       71,    40,    30,    40,   71;

%e      253,   132,    77,    77,  132,  253;

%e      925,   469,   238,   168,  238,  469, 925;

%e     3433,  1724,   828,   450,  450,  828, 1724,  3433;

%e    12871,  6444,  3048,  1452,  990, 1452, 3048,  6444, 12871;

%e    48621, 24320, 11495,  5225, 2717, 2717, 5225, 11495, 24320, 48621;

%e   184757, 92389, 43824, 19734, 9009, 6006, 9009, 19734, 43824, 92389, 184757;

%t T[n_, k_] = Binomial[n+k, k] + Binomial[2*n-k, n-k];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

%o (Magma)

%o T:= func< n,k | Binomial(n+k,n) + Binomial(2*n-k,n) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 29 2021

%o (Sage)

%o def T(n, k): return binomial(n+k,n) + binomial(2*n-k,n)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 29 2021

%Y Row sums are A000984(n+1).

%Y Cf. A001700, A007318, A054142, A085478.

%K nonn,tabl,easy

%O 0,1

%A _Roger L. Bagula_, Dec 19 2009

%E Formula and row sums reference added by the Assoc. Editors of the OEIS, Feb 24 2010