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
KEYWORD
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