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A335050 Array read by descending antidiagonals, T(n,k) is the number of nodes in the pill tree with initial conditions (n,k), for n and k >= 0.

%I A335050 #9 Dec 21 2024 02:13:03
%S A335050 1,2,3,3,7,8,4,12,21,22,5,18,40,63,64,6,25,66,130,195,196,7,33,100,
%T A335050 231,427,624,625,8,42,143,375,803,1428,2054,2055,9,52,196,572,1376,
%U A335050 2805,4860,6916,6917,10,63,260,833,2210,5016,9877,16794,23712,23713
%N A335050 Array read by descending antidiagonals, T(n,k) is the number of nodes in the pill tree with initial conditions (n,k), for n and k >= 0.
%C A335050 See Bayer and Brandt for a description of the pill tree.
%H A335050 Margaret Bayer and Keith Brandt, <a href="http://bayer.faculty.ku.edu/pub/preprints/pill.pdf">The Pill Problem, Lattice Paths and Catalan Numbers</a>, preprint, Mathematics Magazine, Vol. 87, No. 5 (December 2014), pp. 388-394.
%H A335050 Keith Brandt and Kaleb Waite, <a href="https://dl.acm.org/doi/10.5555/1516595.1516624">Using recursion to solve the pill problem</a>, Journal of Computing Sciences in Colleges, Volume 24, Issue 5, May 2009.
%H A335050 Charlotte A. C. Brennan and Helmut Prodinger, <a href="http://math.sun.ac.za/hproding/pdffiles/pillspaper.pdf">The pills problem revisited</a>, preprint, Quaest. Math., 26(4):427-439, 2003.
%H A335050 Donald E. Knuth, John McCarthy, Walter Stromquist, Daniel H. Wagner, and Tim Hesterberg, <a href="https://www.jstor.org/stable/2325015">Problem E3429. Big pills and little pills</a>, The American Mathematical Monthly, 99(7):684, 1992.
%F A335050 T(0,k) = k+1; T(n,0) = 1 + T(n-1,1); T(n,k) = 1 + T(n-1,k+1) + T(n,k-1) for n and k > 0.
%F A335050 T(n,k) = Sum_{j=0..n} (binomial(2*j+k+2, j+1) - binomial(2*j+k+2, j)).
%e A335050 The array begins:
%e A335050     1    2    3    4     5     6 ...
%e A335050     3    7   12   18    25    33 ...
%e A335050     8   21   40   66   100   143 ...
%e A335050    22   63  130  231   375   572 ...
%e A335050    64  195  427  803  1376  2210 ...
%e A335050   196  624 1428 2805  5016  8398 ...
%e A335050   ...
%o A335050 (PARI) T(n,k) = sum(j=0, n, binomial(2*j+k+2, j+1) - binomial(2*j+k+2, j));
%Y A335050 Cf. A000108, A014138 (column 1), A120304 (column 2).
%Y A335050 Cf. A002057 (first differences of column 3).
%K A335050 nonn,tabl
%O A335050 0,2
%A A335050 _Michel Marcus_, May 21 2020