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%I #21 Apr 07 2016 11:35:45
%S 2,4,3,7,9,4,11,22,16,5,16,46,50,25,6,22,86,130,95,36,7,29,148,296,
%T 295,161,49,8,37,239,610,791,581,252,64,9,46,367,1163,1897,1792,1036,
%U 372,81,10,56,541,2083,4166,4900,3612,1716,525,100,11,67,771,3544,8518,12174,11088,6672,2685,715,121,12
%N T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.
%C Table starts
%C ..2...4...7...11....16.....22.....29......37......46.......56.......67
%C ..3...9..22...46....86....148....239.....367.....541......771.....1068
%C ..4..16..50..130...296....610...1163....2083....3544.....5776.....9076
%C ..5..25..95..295...791...1897...4166....8518...16414....30086....52834
%C ..6..36.161..581..1792...4900..12174...27966...60172...122464...237590
%C ..7..49.252.1036..3612..11088..30738...78354..186142...416394...884236
%C ..8..64.372.1716..6672..22716..69498..194634..505912..1233584..2845492
%C ..9..81.525.2685.11517..43065.144111..439791.1241383..3276559..8157227
%C .10.100.715.4015.18832..76714.278707..920491.2803658..7963384.21280337
%C .11.121.946.5786.29458.129844.508937.1808521.5911763.17978389.51325352
%C From _Charles A. Lane_, Aug 22 2013: (Start)
%C The first column is also the coefficients of a in y''[x] - a*x^n*y[x] + b*en*y[x] = 0 where n = 0. The recursion yields coefficients of a, a*b*en, a*b^2*en^2 etc.
%C The second column is obtained when n=1, the third column when n=2. The final column is for n=10.
%C Example: Write a normal recursion for n=4. For convenience set x to 1. Running the recursion yields
%C 1-(b en)/2+(b^2 en^2)/24+1/30 (a-(b^3 en^3)/24)+(-384 a b en+b^4 en^4)/40320+(2064 a b^2 en^2-b^5 en^5)/3628800+(120960 a^2-7104 a b^3 en^3+b^6 en^6)/479001600+(-4682880 a^2 b en+18984 a b^4 en^4-b^7 en^7)/87178291200+(54268416 a^2 b^2 en^2-43008 a b^5 en^5+b^8 en^8)/20922789888000.
%C The coefficient of a is 24, the coefficient of a b en is 384 and the coefficient of a b^2 en^2 is 2064. Dividing by 4! yields a sequence of 1,16,86... , the same as column 5 without the leading 1. There is a hint of unity among the oscillators. (End)
%H R. H. Hardin, <a href="/A225010/b225010.txt">Table of n, a(n) for n = 1..5304</a>
%F Empirical: columns k=1..7 are polynomials of degree k.
%F Empirical: rows n=1..7 are polynomials of degree 2n.
%F T(n,k) = Sum_{j=0..n} C(k+2*j-1,2*j). - _Alois P. Heinz_, Sep 22 2013
%e Some solutions for n=3 k=4
%e ..0..0..0..0....0..1..0..0....0..0..0..0....1..1..1..1....0..0..0..0
%e ..0..0..0..0....0..1..1..0....0..0..0..0....1..1..1..1....1..1..0..0
%e ..0..0..0..1....1..1..1..0....1..1..0..0....1..1..1..1....1..1..1..1
%p T:= (n, k)-> add(binomial(k+2*j-1, 2*j), j=0..n):
%p seq(seq(T(n, 1+d-n), n=1..d), d=1..12); # _Alois P. Heinz_, Sep 22 2013
%t T[n_, k_] := Sum[Binomial[k + 2*j - 1, 2*j], {j, 0, n}]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Apr 07 2016, after _Alois P. Heinz_ *)
%Y Column 2 is A000290(n+1).
%Y Column 3 is A002412(n+1).
%Y Column 4 is A006324(n+1).
%Y Row 1 is A000124.
%Y Row 2 is A223718.
%Y Row 3 is A223659.
%Y Cf. A071920, A071921 (larger and reflected versions of table). - _Alois P. Heinz_, Sep 22 2013
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Apr 23 2013
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