%I #3 Mar 30 2012 18:37:07

%S 1,1,1,3,2,1,15,8,3,1,105,48,15,4,1,945,384,105,23,5,1,10395,3840,945,

%T 176,33,6,1,135135,46080,10395,1689,279,44,7,1,2027025,645120,135135,

%U 19524,2895,400,57,8,1,34459425,10321920,2027025,264207,35685,4384,561

%N Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+2)^2/4 - 1] for m>=0 and then taking partial sums, starting with all 1's in row 0.

%C This is the double factorial analog of Moessner's factorial array (A125714). Compare to triangle A135877 which is generated by a complementary process. A very interesting variant is square array A135878.

%F T(n,0) = (2n)!/n!/2^n; T(n,1) = 2^n*n!; T(n,2) = (2n+1)!/n!/2^n; T(n,3) = A004041(n) = (2n+1)!/n!/2^n * Sum_{k=0..n} 1/(2k+1). T(n,4) = A129890(n) = 2^(n+1)*(n+1)! - (2n+1)!/n!/2^n = T(n+1,1)-T(n+1,0).

%e Square array begins:

%e (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),...;

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

%e (3),(8),15,(23),33,(44),57,71,(86),103,121,(140),161,183,206,(230),..;

%e (15),(48),105,(176),279,(400),561,744,(950),1206,1489,(1800),2171,..;

%e (105),(384),945,(1689),2895,(4384),6555,9129,(12139),16161,20763,..;

%e (945),(3840),10395,(19524),35685,(56448),89055,129072,(177331),245778,...;

%e (10395),(46080),135135,(264207),509985,(836352),1381905,2071215,(2924172),.;

%e (135135),(645120),2027025,(4098240),8294895,(14026752),24137505,...; ...

%e where terms in parenthesis are removed before taking partial sums.

%e For example, to generate row 2 from row 1, remove terms at positions

%e {[(m+2)^2/4-1], m>=0} = [0,1,3,5,8,11,15,19,24,29,35,...] to obtain:

%e [3, 5, 7,8, 10,11, 13,14,15, 17,18,19, 21,22,23,24, 25,26,27,28, ...]

%e then take partial sums to get row 2:

%e [3, 8, 15,23, 33,44, 57,71,86, 103,121,140, 161,183,206,230, ...].

%e Repeating this process will generate all the rows of the triangle,

%e where column 0 will be the odd double factorials (A001147)

%e and column 1 will be the even double factorials (A000165).

%o (PARI) {T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+2)^2/4)-1, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

%Y Cf. columns: A001147, A000165, A004041, A129890; variants: A135878, A125714.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Dec 14 2007