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A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,... and then taking partial sums from right to left, starting with a single 1 in row 0.

%I A135879 #3 Mar 30 2012 18:37:07
%S A135879 1,1,1,2,2,1,1,6,6,4,4,2,2,1,25,25,19,19,13,13,9,5,5,3,1,1,138,138,
%T A135879 113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1,970,970,832,832,694,
%U A135879 694,581,468,468,380,292,292,223,173,123,123,86,62,38,38,23,13,8,3,3,1,8390
%N A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,... and then taking partial sums from right to left, starting with a single 1 in row 0.
%C A135879 Column 0 is A135881 which equals column 0 of square array A135878 and also equals column 0 of triangle A135880. Compare to square array A135878, which is generated by a complementary process. An interesting variant is triangle A135877 in which column 0 equals the double factorials (A001147).
%e A135879 Triangle begins:
%e A135879 1;
%e A135879 1, 1;
%e A135879 2, 2, 1, 1;
%e A135879 6, 6, 4, 4, 2, 2, 1;
%e A135879 25, 25, 19, 19, 13, 13, 9, 5, 5, 3, 1, 1;
%e A135879 138, 138, 113, 113, 88, 88, 69, 50, 50, 37, 24, 24, 15, 10, 5, 5, 2, 1;
%e A135879 970, 970, 832, 832, 694, 694, 581, 468, 468, 380, 292, 292, 223, 173, 123, 123, 86, 62, 38, 38, 23, 13, 8, 3, 3, 1;
%e A135879 8390, 8390, 7420, 7420, 6450, 6450, 5618, 4786, 4786, 4092, 3398, 3398, 2817, 2349, 1881, 1881, 1501, 1209, 917, 917, 694, 521, 398, 275, 275, 189, 127, 89, 51, 51, 28, 15, 7, 4, 1, 1;
%e A135879 There are A135901(n) number of terms in row n.
%e A135879 To generate the triangle, start with a single 1 in row 0,
%e A135879 and then obtain row n+1 from row n by inserting zeros at
%e A135879 positions {[(m+3)^2/4 - 2], m=0,1,2,...} and then
%e A135879 taking reverse partial sums (i.e., summing from right to left).
%e A135879 Start with row 0, insert a zero in front of the '1' at position 0:
%e A135879 [0,1];
%e A135879 take reverse partial sums to get row 1:
%e A135879 [1,1];
%e A135879 insert zeros at positions [0,2]:
%e A135879 [0,1,0,1];
%e A135879 take reverse partial sums to get row 2:
%e A135879 [2,2,1,1];
%e A135879 insert zeros at positions [0,2,4]:
%e A135879 [0,2,0,2,0,1,1];
%e A135879 take reverse partial sums to get row 3:
%e A135879 [6,6,4,4,2,2,1];
%e A135879 insert zeros at positions [0,2,4,7]:
%e A135879 [0,6,0,6,0,4,4,0,2,2,0,1];
%e A135879 take reverse partial sums to get row 4:
%e A135879 [25,25,19,19,13,13,9,5,5,3,1,1];
%e A135879 insert zeros at positions [0,2,4,7,10,14]:
%e A135879 [0,25,0,25,0,19,19,0,13,13,0,9,5,5,0,3,1,1];
%e A135879 take reverse partial sums to get row 5:
%e A135879 [138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1].
%e A135879 Triangle A135880 begins:
%e A135879 1;
%e A135879 1, 1;
%e A135879 2, 2, 1;
%e A135879 6, 7, 3, 1;
%e A135879 25, 34, 15, 4, 1;
%e A135879 138, 215, 99, 26, 5, 1;
%e A135879 970, 1698, 814, 216, 40, 6, 1; ...
%e A135879 and is generated by matrix powers of itself.
%o A135879 (PARI) {T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[]; for(j=1,#A,if(j+m-1==floor((m+2)^2/4)-1,m+=1;B=concat(B,0));B=concat(B,A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));if(k+1>#A,0,A[k+1])}
%Y A135879 Cf. A135881, A135901, A135878, A135880; variants: A135877, A127452, A125781.
%K A135879 nonn,tabl
%O A135879 0,4
%A A135879 _Paul D. Hanna_, Dec 14 2007