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

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, 14, 28, 42, 42, 28, 14, 8, 9, 16, 36, 64, 78, 64, 36, 16, 9, 10, 18, 45, 93, 135, 135, 93, 45, 18, 10, 11, 20, 55, 130, 220, 262, 220, 130, 55, 20, 11, 12, 22, 66

Offset: 0

Author

Gary W. Adamson, Jul 29 2007

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			First few rows of the triangle are:
   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,  14,  28,  42,  42,  28,  14,   8;
   9,  16,  36,  64,  78,  64,  36,  16,   9;
  10,  18,  45,  93, 135, 135,  93,  45,  18,  10;
  ...
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5049

Crossrefs

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

Programs

GAP

a:=Flat(List([0..10],n->List([0..n],k->Binomial(n,k)+n))); # Muniru A Asiru, Jul 16 2018

Magma

/* As triangle */ [[Binomial(n, k) + n: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 17 2018

Mathematica

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 *)

PARI

T(n,k) = binomial(n,k) + n \\ Charles R Greathouse IV, Oct 16 2013

Formula

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;...).

Extensions

Edited, changing formula by Roger L. Bagula, Jul 30 2008

New name from Franklin T. Adams-Watters, Oct 16 2013

Terms 54 onwards from Muniru A Asiru, Jul 16 2018