A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.

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, 1024, 576, 320, 176, 96, 52, 28, 15, 8, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 9, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 10

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

Rows include (essentially) A001787, A001792, A034007, A045623, A045891.

Diagonals include (essentially) A001477, A005408, A008586, A008598, A017113.

Column sums are A058877.

Cf. A111297, A159694, A159695, A159696, A159697. - Philippe Deléham, Apr 21 2009

Cf. A058877, A073371, A130129, A152920.

Sequence in context: A065367 A333780 A033882 * A181596 A182142 A033881

Adjacent sequences: A062108 A062109 A062110 * A062112 A062113 A062114

Keyword

nonn,tabl

Author

Henry Bottomley, May 30 2001

Status

approved