cp's OEIS Frontend

A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

%I A135878 #3 Mar 30 2012 18:37:07
%S A135878 1,1,1,2,2,1,6,6,3,1,25,25,12,4,1,138,138,63,19,5,1,970,970,421,113,
%T A135878 28,6,1,8390,8390,3472,832,190,38,7,1,86796,86796,34380,7420,1560,283,
%U A135878 50,8,1,1049546,1049546,399463,78406,15250,2502,411,63,9,1,14563135,14563135
%N A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.
%C A135878 Column 0 is A135881 which equals column 0 of triangle A135879 and also equals column 0 of triangle A135880. Compare to triangle A135879, which is generated by a complementary process. An interesting variant is square array A135876, in which column 0 equals the double factorials (A001147).
%e A135878 Square array begins:
%e A135878 (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 A135878 (1),2,(3),4,(5),6,7,(8),9,10,(11),12,13,14,(15),16,17,18,(19),20,...;
%e A135878 (2),6,(12),19,(28),38,50,(63),77,93,(110),128,148,169,(191),214,...;
%e A135878 (6),25,(63),113,(190),283,411,(559),728,942,(1181),1446,1766,2116,...;
%e A135878 (25),138,(421),832,(1560),2502,3948,(5714),7830,10740,(14130),18036,...;
%e A135878 (138),970,(3472),7420,(15250),25990,44026,(67112),95918,138343,(189598),..;
%e A135878 (970),8390,(34380),78406,(174324),312667,(563287),897471,1329234,2003240,..;
%e A135878 (8390),86796,(399463),962750,(2291984),4295224,8168819,(13523882),20656067,.;
%e A135878 (86796),1049546,(5344770),13513589,(34169656),66534382,132787852,(227380975),.;
%e A135878 (1049546),14563135,(81097517),213885369,(570682050),1149537869,2395865161,..;
%e A135878 (14563135),228448504,(1377986373),3773851534,(10568874312),21945438536,...;
%e A135878 where terms in parenthesis are removed before taking partial sums.
%e A135878 For example, to generate row 2 from row 1, remove terms at positions
%e A135878 {[(m+3)^2/4-2], m>=0} = [0,2,4,7,10,14,18,23,28,34,...] to obtain:
%e A135878 [2, 4, 6,7, 9,10, 12,13,14, 16,17,18, 20,21,22,23, ...]
%e A135878 then take partial sums to get row 2:
%e A135878 [2, 6, 12,19, 28,38, 50,63,77, 93,110,128, 148,169,191,214, ...].
%e A135878 Repeating this process will generate all the rows of the triangle.
%e A135878 Triangle A135880 begins:
%e A135878 1;
%e A135878 1, 1;
%e A135878 2, 2, 1;
%e A135878 6, 7, 3, 1;
%e A135878 25, 34, 15, 4, 1;
%e A135878 138, 215, 99, 26, 5, 1;
%e A135878 970, 1698, 814, 216, 40, 6, 1; ...
%e A135878 and is generated by matrix powers of itself.
%o A135878 (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+3)^2/4)-2, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}
%Y A135878 Cf. A135881, A135879, A135880; variants: A135876, A125714.
%K A135878 nonn,tabl
%O A135878 0,4
%A A135878 _Paul D. Hanna_, Dec 14 2007