A212362 Triangle by rows, binomial transform of the beheaded Pascal's triangle A074909.

1, 2, 2, 4, 7, 3, 8, 19, 15, 4, 16, 47, 52, 26, 5, 32, 111, 155, 110, 40, 6, 64, 255, 426, 385, 200, 57, 7, 128, 575, 1113, 1211, 805, 329, 77, 8, 256, 1279, 2808, 3556, 2856, 1498, 504, 100, 9, 512, 2815, 6903, 9948, 9324, 5922, 2562, 732, 126, 10

Offset: 0

Author

Gary W. Adamson, Jun 29 2012

Keywords

• nonn
• tabl

Comments

Row sums of the triangle inverse = A027641/A027642, the Bernoulli numbers; (1, -1/2, 1/6, 0, -1/30,...)

Examples

			First few rows of the triangle are:
    1;
    2,    2;
    4,    7,    3;
    8,   19,   15,    4
   16,   47,   52,   26,    5;
   32,  111,  155,  110,   40,    6;
   64,  255,  426,  385,  200,   57,   7;
  128,  575, 1113, 1211,  805,  329,  77,   8;
  256, 1279, 2808, 3556, 2856, 1498, 504, 100, 9;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A074909, A027641/A027642, A027649 (row sums), A006589 (2nd column), A106515.

Programs

Magma

A074909:= func< n,k | k lt 0 or k gt n select 0 else Binomial(n+1, k) >;
A212362:= func< n,k | (&+[ Binomial(n,j)*A074909(j, k) : j in [0..n]]) >;
[A212362(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 05 2021

Maple

A212362 := proc(n,k)
        add( binomial(n,i)*A074909(i,k),i=0..n) ;
end proc: # R. J. Mathar, Aug 03 2015

Mathematica

T[n_, k_]= 2^(n-k)*Binomial[n+1, k] + (2^(n-k) -1)*Binomial[n, k-1];
Table[T[n, k] , {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, Aug 05 2021 *)

Sage

def T(n, k): return 2^(n-k)*binomial(n+1, k) + (2^(n-k) - 1)*binomial(n, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 05 2021

Formula

Binomial transform of the beheaded Pascal's triangle (A074909) as a matrix. (The beheaded Pascal matrix deletes the rightmost border of 1's.)

From G. C. Greubel, Aug 05 2021: (Start)

T(n, k) = Sum_{j=0..n} binomial(n, j)*binomial(j+1, k) - binomial(n, k-1), with T(n, 0) = 2^n.

T(n, k) = 2^(n-k)*binomial(n+1, k) + (2^(n-k) - 1)*binomial(n, k-1).

Sum_{k=0..n} T(n, k) = A027649(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A106515(n). (End)

Extensions

a(22) corrected by G. C. Greubel, Aug 05 2021