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

%I A062111 #21 Oct 01 2022 00:19:52
%S A062111 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 A062111 11,6,448,256,144,80,44,24,13,7,1024,576,320,176,96,52,28,15,8,2304,
%U A062111 1280,704,384,208,112,60,32,17,9,5120,2816,1536,832,448,240,128,68,36,19,10
%N A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.
%C A062111 From _Philippe Deléham_, Apr 15 2007: (Start)
%C A062111 This triangle can be found in the Laisant reference in the following form:
%C A062111   .......................5...11..
%C A062111   ...................4...9...20..
%C A062111   ...............3...7..16...36..
%C A062111   ...........2...5..12..28.......
%C A062111   .......1...3...8..20..48.......
%C A062111   ...0...1...4..12..32..80....... (End)
%C A062111 Triangle A152920 reversed. - _Philippe Deléham_, Apr 21 2009
%H A062111 G. C. Greubel, <a href="/A062111/b062111.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A062111 F. Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>
%H A062111 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 A062111 A(n, k) = A(n, k-1) + A(n+1, k) if k > n with A(n, n) = n.
%F A062111 A(n, k) = (k+n)*2^(k-n-1) if k >= n.
%F A062111 T(2*n, n) = 3*n*2^(n-1) = 3*A001787(n). - _Philippe Deléham_, Apr 21 2009
%F A062111 From _G. C. Greubel_, Sep 28 2022: (Start)
%F A062111 T(n, k) = 2^(n-k-1)*(n+k) for 0 <= k <= n, n >= 0.
%F A062111 T(m*n, n) = 2^((m-1)*n-1)*(m+1)*A001477(n), m >= 1.
%F A062111 T(2*n-1, n-1) = A130129(n-1).
%F A062111 T(2*n+1, n-1) = 12*A001787(n).
%F A062111 Sum_{k=0..n} T(n, k) = A058877(n+1).
%F A062111 Sum_{k=0..n} (-1)^k*T(n, k) = 3*A073371(n-2), n >= 2.
%F A062111 T(n, k) = A152920(n, n-k). (End)
%e A062111 As a lower triangle (T(n, k)):
%e A062111     0;
%e A062111     1,   1;
%e A062111     4,   3,   2;
%e A062111    12,   8,   5,  3;
%e A062111    32,  20,  12,  7,  4;
%e A062111    80,  48,  28, 16,  9,  5;
%e A062111   192, 112,  64, 36, 20, 11,  6;
%e A062111   448, 256, 144, 80, 44, 24, 13, 7;
%t A062111 Table[2^(n-k-1)*(n+k), {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 28 2022 *)
%o A062111 (Magma) [2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 28 2022
%o A062111 (SageMath)
%o A062111 def A062111(n,k): return 2^(n-k-1)*(n+k)
%o A062111 flatten([[A062111(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Sep 28 2022
%Y A062111 Rows include (essentially) A001787, A001792, A034007, A045623, A045891.
%Y A062111 Diagonals include (essentially) A001477, A005408, A008586, A008598, A017113.
%Y A062111 Column sums are A058877.
%Y A062111 Cf. A111297, A159694, A159695, A159696, A159697. - _Philippe Deléham_, Apr 21 2009
%Y A062111 Cf. A058877, A073371, A130129, A152920.
%K A062111 nonn,tabl
%O A062111 0,4
%A A062111 _Henry Bottomley_, May 30 2001