cp's OEIS Frontend

A135876 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.

%I A135876 #3 Mar 30 2012 18:37:07
%S A135876 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 A135876 176,33,6,1,135135,46080,10395,1689,279,44,7,1,2027025,645120,135135,
%U A135876 19524,2895,400,57,8,1,34459425,10321920,2027025,264207,35685,4384,561
%N A135876 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 A135876 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 A135876 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 A135876 Square array begins:
%e A135876 (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 A135876 (1),(2),3,(4),5,(6),7,8,(9),10,11,(12),13,14,15,(16),17,18,19,(20),...;
%e A135876 (3),(8),15,(23),33,(44),57,71,(86),103,121,(140),161,183,206,(230),..;
%e A135876 (15),(48),105,(176),279,(400),561,744,(950),1206,1489,(1800),2171,..;
%e A135876 (105),(384),945,(1689),2895,(4384),6555,9129,(12139),16161,20763,..;
%e A135876 (945),(3840),10395,(19524),35685,(56448),89055,129072,(177331),245778,...;
%e A135876 (10395),(46080),135135,(264207),509985,(836352),1381905,2071215,(2924172),.;
%e A135876 (135135),(645120),2027025,(4098240),8294895,(14026752),24137505,...; ...
%e A135876 where terms in parenthesis are removed before taking partial sums.
%e A135876 For example, to generate row 2 from row 1, remove terms at positions
%e A135876 {[(m+2)^2/4-1], m>=0} = [0,1,3,5,8,11,15,19,24,29,35,...] to obtain:
%e A135876 [3, 5, 7,8, 10,11, 13,14,15, 17,18,19, 21,22,23,24, 25,26,27,28, ...]
%e A135876 then take partial sums to get row 2:
%e A135876 [3, 8, 15,23, 33,44, 57,71,86, 103,121,140, 161,183,206,230, ...].
%e A135876 Repeating this process will generate all the rows of the triangle,
%e A135876 where column 0 will be the odd double factorials (A001147)
%e A135876 and column 1 will be the even double factorials (A000165).
%o A135876 (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 A135876 Cf. columns: A001147, A000165, A004041, A129890; variants: A135878, A125714.
%K A135876 nonn,tabl
%O A135876 0,4
%A A135876 _Paul D. Hanna_, Dec 14 2007