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C.-A. Laisant, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k201179s/f211.image">Sur les tableaux de sommes - Nouvelles applications</a>, Compt. Rendus de l'Association Francaise pour l'Avancement des Sciences, Aout. 04, 1893, pp. 206-216 (table given on p. 212).

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........ .......................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)

G. C. Greubel, <a href="/A062111/b062111.txt">Rows n = 0..50 of the triangle, flattened</a>

a(n, n) = n; aA(n, k) = aA(n, k-1) + aA(n+1, k) if k > n with A(n, n) = n.

aA(n, k) = (k+n)*2^(k-n-1) if k >= n.

T(2n,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)

Rows start (0,1,4,12,32,...), (1,3,8,20,...), (2,5,12,...), (3,7,...), etc.

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;

Table[2^(n-k-1)*(n+k), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 28 2022 *)

(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

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

Rows include (essentially) A001787, A001792, A045623, A045891, A034007. Diagonals include (essentially) A001477, A005408, A008586, A017113, A017113, A008598. Column sums are A058877, A017113.

Column sums are A058877.

Cf. A058877, A073371, A130129, A152920.

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From Philippe Deléham, Apr 15 2007 : (Start)

Triangle A152920 reversed . [_- _Philippe Deléham_, Apr 21 2009]

a(n, n) = n; a(n, k) = a(n, k-1) + a(n+1, k) if k>n. a(n, k)=(k+n)*2^(k-n-1) if k >= n.

a(n, k) = (k+n)*2^(k-n-1) if k >= n.

T(2n,n) = 3*n*2^(n-1) = 3*A001787(n). [_- _Philippe Deléham_, Apr 21 2009]

Cf. A111297, A159694, A159695, A159696, A159697. [_- _Philippe Deléham_, Apr 21 2009]

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