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Table read by rows: prime power factors of central binomial coefficients, cf. A000984.
9

%I #17 Jan 21 2016 11:23:40

%S 1,2,2,3,4,5,2,5,7,4,9,7,4,3,7,11,8,3,11,13,2,9,5,11,13,4,5,11,13,17,

%T 4,11,13,17,19,8,3,7,13,17,19,4,7,13,17,19,23,8,25,7,17,19,23,8,27,25,

%U 17,19,23,16,9,5,17,19,23,29,2,9,5,17,19,23,29,31

%N Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

%C T(n,k) = A141809(A000984(n)),k) for k = 0..A067434(n)-1.

%H Reinhard Zumkeller, <a href="/A226078/b226078.txt">Rows n = 0..250 of triangle, flattened</a>

%e . n initial rows A000984(n) A226047(n)

%e . ---+------------------------------+-------------+------------

%e . 0 [1] 1

%e . 1 [2] 2 2

%e . 2 [2,3] 6 3

%e . 3 [4,5] 20 5

%e . 4 [2,5,7] 70 7

%e . 5 [4,9,7] 252 9

%e . 6 [4,3,7,11] 924 11

%e . 7 [8,3,11,13] 3432 13

%e . 8 [2,9,5,11,13] 12870 13

%e . 9 [4,5,11,13,17] 48620 17

%e . 10 [4,11,13,17,19] 184756 19

%e . 11 [8,3,7,13,17,19] 705432 19

%e . 12 [4,7,13,17,19,23] 2704156 23

%e . 13 [8,25,7,17,19,23] 10400600 25

%e . 14 [8,27,25,17,19,23] 40116600 27

%e . 15 [16,9,5,17,19,23,29] 155117520 29

%e . 16 [2,9,5,17,19,23,29,31] 601080390 31

%e . 17 [4,27,5,11,19,23,29,31] 2333606220 31

%e . 18 [4,3,25,7,11,19,23,29,31] 9075135300 31

%e . 19 [8,3,25,7,11,23,29,31,37] 35345263800 37

%e . 20 [4,9,5,7,11,13,23,29,31,37] 137846528820 37 .

%p f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):

%p b:= proc(n) local p;

%p p:= add(f(n+i) -f(i), i=1..n);

%p seq(`if`(coeff(p, x, i)>0,

%p i^coeff(p, x, i), NULL), i=1..degree(p))

%p end:

%p T:= n-> `if`(n=0, 1, b(n)):

%p seq(T(n), n=0..30); # _Alois P. Heinz_, May 25 2013

%t Table[Power @@@ FactorInteger[(2n)!/n!^2] , {n, 0, 30}] // Flatten (* _Jean-François Alcover_, Jul 29 2015 *)

%o (Haskell)

%o a226078 n k = a226078_tabf !! n !! k

%o a226078_row n = a226078_tabf !! n

%o a226078_tabf = map a141809_row a000984_list

%Y Cf. A067434 (row lengths), A001316 (left edge), A060308 (right edge), A226047 (row maxima), A226083 (row minima), A000984 (row products).

%Y Cf. A267823.

%K nonn,tabf,look

%O 0,2

%A _Reinhard Zumkeller_, May 25 2013