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%I #35 Nov 03 2021 22:21:44
%S 2,4,10,6,14,50,8,22,70,230,12,26,98,250,1150,16,34,110,290,1250,5050,
%T 18,38,130,310,1450,5150,22310,20,44,154,370,1550,5290,23230,106030,
%U 24,46,170,406,1850,5350,23690,106490,510050,28,52,182,410,2030,5450,24610,107410,513130,2065450
%N Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.
%C This is a permutation of the positive even numbers (A299174).
%C The union of all odd-indexed columns gives A319796, the even numbers in A071562.
%C The union of all even-indexed columns gives A319802, the even numbers in A071561.
%e From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
%e The 10x10 section of table T(n,k):
%e (Table with first 20 terms from _Omar E. Pol_)
%e ------------------------------------------------------------------
%e n\k | 1 2 3 4 5 6 7 8 9 10 ...
%e ------------------------------------------------------------------
%e 1 | 2 10 50 230 1150 5050 22310 106030 510050 2065450
%e 2 | 4 14 70 250 1250 5150 23230 106490 513130 2115950
%e 3 | 6 22 98 290 1450 5290 23690 107410 520150 2126050
%e 4 | 8 26 110 310 1550 5350 24610 110170 530150 2157850
%e 5 | 12 34 130 370 1850 5450 25070 112010 530450 2164070
%e 6 | 16 38 154 406 2030 5650 25250 112930 532450 2168150
%e 7 | 18 44 170 410 2050 5750 25750 114770 534290 2176550
%e 8 | 20 46 182 430 2150 6250 25990 115690 537050 2186650
%e 9 | 24 52 190 434 2170 6350 26450 116150 540350 2216950
%e 10| 28 58 238 470 2350 6550 26750 117070 544870 2219650
%e ... (End)
%t (* function a341969 is defined in A341969 *)
%t sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]<pMax, AppendTo[list[[p]], i]]]; list]
%t rank[n_] := n-row[n-1](row[n-1]+1)/2
%t parts[n_] := row[n-1]-rank[n]+2
%t a320537[sMatrix_, aD_] := Map[sMatrix[[rank[#], parts[#]]]&, Range[aD (aD+1)/2]]/; MatrixQ[sMatrix]&&aD<=Length[sMatrix]
%t m2500000=sArray[2500000, 10] (* entire 10x10 matrix needs to be computed *)
%t a320537[m2500000, 10] (* Sequence Data a(1..55) *)
%t (* _Hartmut F. W. Hoft_, Oct 06 2021 *)
%Y Row 1 is A320521.
%Y Column 1 gives A174973 = A238443, without the 1.
%Y Column 2 gives A244894.
%Y Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.
%K nonn,tabl
%O 1,1
%A _Omar E. Pol_, Oct 15 2018
%E Terms a(21) and beyond from _Hartmut F. W. Hoft_, Oct 06 2021