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

0,4

Links

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

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)

Example

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

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A342552 A078908 A159797 * A361442 A288778 A290139

Adjacent sequences: A152917 A152918 A152919 * A152921 A152922 A152923

Keyword

nonn,tabl,easy

Author

Paul Curtz, Dec 15 2008

Extensions

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

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

Status

approved