OFFSET
0,4
COMMENTS
From Philippe Deléham, Apr 15 2007: (Start)
This triangle can be found in the Laisant reference in the following form:
.......................5...11..
...................4...9...20..
...............3...7..16...36..
...........2...5..12..28.......
.......1...3...8..20..48.......
...0...1...4..12..32..80....... (End)
Triangle A152920 reversed. - Philippe Deléham, Apr 21 2009
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
F. Ellermann, Illustration of binomial transforms
C.-A. Laisant, Sur les tableaux de sommes - Nouvelles applications, Compt. Rendus de l'Association Francaise pour l'Avancement des Sciences, Aout 04 1893, pp. 206-216 (table given on p. 212).
FORMULA
A(n, k) = A(n, k-1) + A(n+1, k) if k > n with A(n, n) = n.
A(n, k) = (k+n)*2^(k-n-1) if k >= n.
T(2*n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 21 2009
From G. C. Greubel, Sep 28 2022: (Start)
T(n, k) = 2^(n-k-1)*(n+k) for 0 <= k <= n, n >= 0.
T(m*n, n) = 2^((m-1)*n-1)*(m+1)*A001477(n), m >= 1.
T(2*n-1, n-1) = A130129(n-1).
T(2*n+1, n-1) = 12*A001787(n).
Sum_{k=0..n} T(n, k) = A058877(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = 3*A073371(n-2), n >= 2.
T(n, k) = A152920(n, n-k). (End)
EXAMPLE
As a lower triangle (T(n, k)):
0;
1, 1;
4, 3, 2;
12, 8, 5, 3;
32, 20, 12, 7, 4;
80, 48, 28, 16, 9, 5;
192, 112, 64, 36, 20, 11, 6;
448, 256, 144, 80, 44, 24, 13, 7;
MATHEMATICA
Table[2^(n-k-1)*(n+k), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 28 2022 *)
PROG
(Magma) [2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 28 2022
(SageMath)
def A062111(n, k): return 2^(n-k-1)*(n+k)
flatten([[A062111(n, k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 28 2022
CROSSREFS
Column sums are A058877.
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, May 30 2001
STATUS
approved