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

0,2

Comments

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

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A249758 A157470 A120280 * A209142 A368518 A115754

Adjacent sequences: A212359 A212360 A212361 * A212363 A212364 A212365

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 29 2012

Extensions

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

Status

approved