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%I #21 Feb 13 2022 23:17:44
%S 1,2,1,4,3,1,8,8,4,1,16,20,12,5,1,32,48,32,16,6,1,64,112,80,44,20,7,1,
%T 128,256,192,112,56,24,8,1,256,576,448,272,144,68,28,9,1,512,1280,
%U 1024,640,352,176,80,32,10,1,1024,2816,2304,1472,832,432,208,92,36,11,1
%N Binomial transform triangle, read by rows.
%C Let P = Pascal's triangle as an infinite lower triangular matrix and A is the infinite array of arithmetic sequences as shown in A077028:
%C 1, 1, 1, 1, 1, ...
%C 1, 2, 3, 4, 5, ...
%C 1, 3, 5, 7, 9, ...
%C 1, 4, 7, 10, 13, ...
%C 1, 5, 9, 13, 17, ...
%C Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.
%F n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2, ...).
%F From _Peter Bala_, Jul 26 2015: (Start)
%F T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.
%F O.g.f.: (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....
%F k-th column g.f.: (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)
%e Column 3: 1, 5, 16, 44, 112, ... (A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7, ...).
%e Triangle begins:
%e 1;
%e 2, 1;
%e 4, 3, 1;
%e 8, 8, 4, 1;
%e 16, 20, 12, 5, 1;
%e 32, 48, 32, 16, 6, 1;
%e 64, 112, 80, 44, 20, 7, 1;
%e 128, 256, 192, 112, 56, 24, 8, 1;
%e 256, 576, 448, 272, 144, 68, 28, 9, 1;
%e 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1;
%e 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1;
%e ...
%p seq(seq((2 + k*(n - k))*2^(n-k-1),k=0..n),n=0..10); # _Peter Bala_, Jul 26 2015
%t t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* _Vincenzo Librandi_, Jul 26 2015 *)
%o (Magma) /* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 26 2015
%Y Cf. A077028, A001792, A001787, A053220, A016777, A014480.
%K nonn,tabl
%O 0,2
%A _Gary W. Adamson_, Apr 23 2005
%E More terms from _Philippe Deléham_, Mar 31 2007
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