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

Author

Henry Bottomley, May 30 2001

Keywords

• nonn
• tabl

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

Examples

			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;
		

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).

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.

Programs

Magma

[2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 28 2022

Mathematica

Table[2^(n-k-1)*(n+k), {n,0,12}, {k,0,n}]//Flatten (* 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

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)