0,4

This is the triple factorial variant of Moessner's factorial array described by A125714 and also of the double factorial array A135876. Another very interesting variant is A136217.

Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.

Square array begins:

(1),(1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,...;

(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),19,20,21,..;

(4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;

(28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;

(280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;

(3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;

(58240),(209440),(524880),1106560,(1864456),3082240,...;

where terms in parenthesis are at positions {[m*(m+5)/6], m>=0}

and are removed before taking partial sums to obtain the next row.

To generate the array, start with all 1's in row 0; from then on,

obtain row n+1 from row n by first removing terms in row n at

positions {[m*(m+5)/6], m>=0} and then taking partial sums.

For example, to generate row 2 from row 1:

[(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),...],

remove terms at positions [0,1,2,4,6,8,11,14,17,...] to get:

[4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]

then take partial sums to obtain row 2:

[4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].

Continuing in this way will generate all the rows of this array.

(PARI) {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+5))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.

Sequence in context: A105623 A158835 A138271 this_sequence A136215 A136737 A004551

Adjacent sequences: A136209 A136210 A136211 this_sequence A136213 A136214 A136215

nice,nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 22 2007