A152920 Triangle read by rows: triangle A062111 reversed.

0, 1, 1, 2, 3, 4, 3, 5, 8, 12, 4, 7, 12, 20, 32, 5, 9, 16, 28, 48, 80, 6, 11, 20, 36, 64, 112, 192, 7, 13, 24, 44, 80, 144, 256, 448, 8, 15, 28, 52, 96, 176, 320, 576, 1024, 9, 17, 32, 60, 112, 208, 384, 704, 1280, 2304, 10, 19, 36, 68, 128, 240, 448, 832, 1536, 2816, 5120

Offset: 0

Author

Paul Curtz, Dec 15 2008

Keywords

• nonn
• tabl
• easy

Examples

			Triangle starts:
  0;
  1,  1;
  2,  3,  4;
  3,  5,  8, 12;
  4,  7, 12, 20, 32;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened

Crossrefs

Cf. A053220, A058877 (row sums), A193844, A212697.

Columns and diagonals: A001787, A001792, A034007, A045623, A045891, A111297, A159694, A159695, A159696, A159697.

Programs

Magma

[2^k*(n-k/2): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2022

Maple

A062111 := proc(n,k) (k+n)*2^(k-n-1) ; end: A152920 := proc(n,k) A062111(n-k,n) ; end: for n from 0 to 15 do for k from 0 to n do printf("%d,",A152920(n,k)) ; od: od: # R. J. Mathar, Jan 22 2009
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k=0, n, T(n, k-1)+T(n-1, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 12 2022

Mathematica

t[0, k_]:= k; t[n_, k_]:= t[n, k]= t[n-1, k] + t[n-1, k+1];
Table[t[n-k, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Jean-François Alcover, Sep 11 2016 *)

SageMath

flatten([[2^(k-1)*(2*n-k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 27 2022

Formula

Row sums: (2^n-1)(n+1) = A058877(n). - R. J. Mathar, Jan 22 2009

T(2n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 20 2009

From Werner Schulte, Jul 31 2020: (Start)

T(n, k) = (2*n-k) * 2^(k-1) for 0 <= k <= n.

G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = t*(1+x-3*x*t) / ((1-t)^2 * (1-2*x*t)^2).

Sum_{k=0..n} (-1)^k * binomial(n,k) * T(n,k) = 0 for n >= 0.

Sum_{k=0..n} binomial(n,k) * T(n,k) = 2*n * 3^(n-1) for n >= 0.

Define the array B(n,p) = (Sum_{k=0..n} binomial(p+k,p) * T(n,k))/(n+p+1) for n >= 0 and p >= 0. Then see the comment of Robert Coquereaux (2014) at A193844. Conjecture: B(n+1,p) = A(n,p). (End)

T(n, k) = T(n, k-1) + T(n-1, k-1) for k>=1, T(n,0) = n. - Alois P. Heinz, Sep 12 2022

From G. C. Greubel, Sep 27 2022: (Start)

T(n, n-1) = A001792(n).

T(2*n-1, n-1) = A053220(n).

T(2*n+1, n-1) = 3*A001792(n).

T(m*n, n) = (2*m-1)*A001787(n), for m >= 1. (End)

Extensions

Edited by N. J. A. Sloane, Dec 19 2008

More terms from R. J. Mathar, Jan 22 2009