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Upper-right triangle resulting from binomial transform calculation for nonnegative integers.
16

%I #21 Oct 01 2022 00:19:52

%S 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,

%T 11,6,448,256,144,80,44,24,13,7,1024,576,320,176,96,52,28,15,8,2304,

%U 1280,704,384,208,112,60,32,17,9,5120,2816,1536,832,448,240,128,68,36,19,10

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

%C From _Philippe Deléham_, Apr 15 2007: (Start)

%C This triangle can be found in the Laisant reference in the following form:

%C .......................5...11..

%C ...................4...9...20..

%C ...............3...7..16...36..

%C ...........2...5..12..28.......

%C .......1...3...8..20..48.......

%C ...0...1...4..12..32..80....... (End)

%C Triangle A152920 reversed. - _Philippe Deléham_, Apr 21 2009

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

%H F. Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>

%H 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).

%F A(n, k) = A(n, k-1) + A(n+1, k) if k > n with A(n, n) = n.

%F A(n, k) = (k+n)*2^(k-n-1) if k >= n.

%F T(2*n, n) = 3*n*2^(n-1) = 3*A001787(n). - _Philippe Deléham_, Apr 21 2009

%F From _G. C. Greubel_, Sep 28 2022: (Start)

%F T(n, k) = 2^(n-k-1)*(n+k) for 0 <= k <= n, n >= 0.

%F T(m*n, n) = 2^((m-1)*n-1)*(m+1)*A001477(n), m >= 1.

%F T(2*n-1, n-1) = A130129(n-1).

%F T(2*n+1, n-1) = 12*A001787(n).

%F Sum_{k=0..n} T(n, k) = A058877(n+1).

%F Sum_{k=0..n} (-1)^k*T(n, k) = 3*A073371(n-2), n >= 2.

%F T(n, k) = A152920(n, n-k). (End)

%e As a lower triangle (T(n, k)):

%e 0;

%e 1, 1;

%e 4, 3, 2;

%e 12, 8, 5, 3;

%e 32, 20, 12, 7, 4;

%e 80, 48, 28, 16, 9, 5;

%e 192, 112, 64, 36, 20, 11, 6;

%e 448, 256, 144, 80, 44, 24, 13, 7;

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

%o (Magma) [2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 28 2022

%o (SageMath)

%o def A062111(n,k): return 2^(n-k-1)*(n+k)

%o flatten([[A062111(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Sep 28 2022

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

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

%Y Column sums are A058877.

%Y Cf. A111297, A159694, A159695, A159696, A159697. - _Philippe Deléham_, Apr 21 2009

%Y Cf. A058877, A073371, A130129, A152920.

%K nonn,tabl

%O 0,4

%A _Henry Bottomley_, May 30 2001