A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

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

Offset: 0

Author

Reinhard Zumkeller, May 25 2013

Keywords

• nonn
• tabf
• look

Examples

			.   n        initial rows               A000984(n)   A226047(n)
.  ---+------------------------------+-------------+------------
.   0   [1]                                      1
.   1   [2]                                      2            2
.   2   [2,3]                                    6            3
.   3   [4,5]                                   20            5
.   4   [2,5,7]                                 70            7
.   5   [4,9,7]                                252            9
.   6   [4,3,7,11]                             924           11
.   7   [8,3,11,13]                           3432           13
.   8   [2,9,5,11,13]                        12870           13
.   9   [4,5,11,13,17]                       48620           17
.  10   [4,11,13,17,19]                     184756           19
.  11   [8,3,7,13,17,19]                    705432           19
.  12   [4,7,13,17,19,23]                  2704156           23
.  13   [8,25,7,17,19,23]                 10400600           25
.  14   [8,27,25,17,19,23]                40116600           27
.  15   [16,9,5,17,19,23,29]             155117520           29
.  16   [2,9,5,17,19,23,29,31]           601080390           31
.  17   [4,27,5,11,19,23,29,31]         2333606220           31
.  18   [4,3,25,7,11,19,23,29,31]       9075135300           31
.  19   [8,3,25,7,11,23,29,31,37]      35345263800           37
.  20   [4,9,5,7,11,13,23,29,31,37]   137846528820           37 .
		

Links

• Reinhard Zumkeller, Rows n = 0..250 of triangle, flattened

Crossrefs

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

Cf. A267823.

Programs

Haskell

a226078 n k = a226078_tabf !! n !! k
a226078_row n = a226078_tabf !! n
a226078_tabf = map a141809_row a000984_list

Maple

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
b:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      seq(`if`(coeff(p, x, i)>0,
             i^coeff(p, x, i), NULL), i=1..degree(p))
    end:
T:= n-> `if`(n=0, 1, b(n)):
seq(T(n), n=0..30);  # Alois P. Heinz, May 25 2013

Mathematica

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

PARI

row(n)= if(n<1, [1], [ e[1]^e[2] |e<-Col(factor(binomial(2*n, n)))]); \\ Ruud H.G. van Tol, Nov 18 2024

Formula

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