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Triangle read by rows: T(n,k) = binomial(n,k) + n.
2

%I #31 May 12 2023 09:29:59

%S 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 14,28,42,42,28,14,8,9,16,36,64,78,64,36,16,9,10,18,45,93,135,135,93,

%U 45,18,10,11,20,55,130,220,262,220,130,55,20,11,12,22,66

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

%C Row sums = A131924: (1, 4, 10, 20, 36, 62, 106, 184, ...).

%H Muniru A Asiru, <a href="/A131923/b131923.txt">Table of n, a(n) for n = 0..5049</a>

%F 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 First few rows of the triangle are:

%e 1;

%e 2, 2;

%e 3, 4, 3;

%e 4, 6, 6, 4;

%e 5, 8, 10, 8, 5;

%e 6, 10, 15, 15, 10, 6;

%e 7, 12, 21, 26, 21, 12, 7;

%e 8, 14, 28, 42, 42, 28, 14, 8;

%e 9, 16, 36, 64, 78, 64, 36, 16, 9;

%e 10, 18, 45, 93, 135, 135, 93, 45, 18, 10;

%e ...

%t 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 (PARI) T(n,k) = binomial(n,k) + n \\ _Charles R Greathouse IV_, Oct 16 2013

%o (GAP) a:=Flat(List([0..10],n->List([0..n],k->Binomial(n,k)+n))); # _Muniru A Asiru_, Jul 16 2018

%o (Magma) /* As triangle */ [[Binomial(n, k) + n: k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 17 2018

%Y Cf. A002024, A007318, A000012, A003056, A131924 (row sums), A003991.

%K nonn,tabl,easy

%O 0,2

%A _Gary W. Adamson_, Jul 29 2007

%E Edited, changing formula by _Roger L. Bagula_, Jul 30 2008

%E New name from _Franklin T. Adams-Watters_, Oct 16 2013

%E Terms 54 onwards from _Muniru A Asiru_, Jul 16 2018