cp's OEIS Frontend

A131923 Triangle read by rows: T(n,k) = binomial(n,k) + n.

%I A131923 #31 May 12 2023 09:29:59
%S A131923 1,2,2,3,4,3,4,6,6,4,5,8,10,8,5,6,10,15,15,10,6,7,12,21,26,21,12,7,8,
%T A131923 14,28,42,42,28,14,8,9,16,36,64,78,64,36,16,9,10,18,45,93,135,135,93,
%U A131923 45,18,10,11,20,55,130,220,262,220,130,55,20,11,12,22,66
%N A131923 Triangle read by rows: T(n,k) = binomial(n,k) + n.
%C A131923 Row sums = A131924: (1, 4, 10, 20, 36, 62, 106, 184, ...).
%H A131923 Muniru A Asiru, <a href="/A131923/b131923.txt">Table of n, a(n) for n = 0..5049</a>
%F A131923 A007318 + A002024 - A000012 = A007318 + A003056 as infinite lower triangular matrices. A002024 = (1; 2,2; 3,3,3;...); A007318 = Pascal's triangle and A000012 = (1; 1,1; 1,1,1;...).
%e A131923 First few rows of the triangle are:
%e A131923    1;
%e A131923    2,   2;
%e A131923    3,   4,   3;
%e A131923    4,   6,   6,   4;
%e A131923    5,   8,  10,   8,   5;
%e A131923    6,  10,  15,  15,  10,   6;
%e A131923    7,  12,  21,  26,  21,  12,   7;
%e A131923    8,  14,  28,  42,  42,  28,  14,   8;
%e A131923    9,  16,  36,  64,  78,  64,  36,  16,   9;
%e A131923   10,  18,  45,  93, 135, 135,  93,  45,  18,  10;
%e A131923   ...
%t A131923 T[n_, m_] = Binomial[n, m] + n; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] (* _Roger L. Bagula_, Jul 30 2008 *)
%o A131923 (PARI) T(n,k) = binomial(n,k) + n \\ _Charles R Greathouse IV_, Oct 16 2013
%o A131923 (GAP) a:=Flat(List([0..10],n->List([0..n],k->Binomial(n,k)+n))); # _Muniru A Asiru_, Jul 16 2018
%o A131923 (Magma) /* As triangle */ [[Binomial(n, k) + n: k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 17 2018
%Y A131923 Cf. A002024, A007318, A000012, A003056, A131924 (row sums), A003991.
%K A131923 nonn,tabl,easy
%O A131923 0,2
%A A131923 _Gary W. Adamson_, Jul 29 2007
%E A131923 Edited, changing formula by _Roger L. Bagula_, Jul 30 2008
%E A131923 New name from _Franklin T. Adams-Watters_, Oct 16 2013
%E A131923 Terms 54 onwards from _Muniru A Asiru_, Jul 16 2018