cp's OEIS Frontend

A135877 Triangle, read by rows of n(n+1)+1 terms, where row n+1 is generated from row n by inserting zeros at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then taking partial sums from right to left, starting with a single 1 in row 0.

%I A135877 #3 Mar 30 2012 18:37:07
%S A135877 1,1,1,1,3,3,3,2,2,1,1,15,15,15,12,12,9,9,6,4,4,2,1,1,105,105,105,90,
%T A135877 90,75,75,60,48,48,36,27,27,18,12,8,8,4,2,1,1,945,945,945,840,840,735,
%U A135877 735,630,540,540,450,375,375,300,240,192,192,144,108,81,81,54,36,24,16,16,8
%N A135877 Triangle, read by rows of n(n+1)+1 terms, where row n+1 is generated from row n by inserting zeros at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then taking partial sums from right to left, starting with a single 1 in row 0.
%C A135877 Compare to square array A135876 which is generated by a complementary process. Compare to triangle A127452 which generates the factorials in column 0. A very interesting variant is triangle A135879.
%F A135877 Column 0 equals the double factorials A001147(n) = (2n)!/(n!*2^n).
%e A135877 Triangle begins:
%e A135877 1;
%e A135877 1, 1, 1;
%e A135877 3, 3, 3, 2, 2, 1, 1;
%e A135877 15, 15, 15, 12, 12, 9, 9, 6, 4, 4, 2, 1, 1;
%e A135877 105, 105, 105, 90, 90, 75, 75, 60, 48, 48, 36, 27, 27, 18, 12, 8, 8, 4, 2, 1, 1;
%e A135877 945, 945, 945, 840, 840, 735, 735, 630, 540, 540, 450, 375, 375, 300, 240, 192, 192, 144, 108, 81, 81, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1; ...
%e A135877 To generate the triangle, start with a single 1 in row 0,
%e A135877 and then obtain row n+1 from row n by inserting zeros
%e A135877 at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then
%e A135877 taking reverse partial sums (i.e., summing from right to left).
%e A135877 Start with row 0, insert 2 zeros in front of the '1':
%e A135877 [0,0,1];
%e A135877 take reverse partial sums to get row 1:
%e A135877 [1,1,1];
%e A135877 insert zeros at positions [0,1,3,5]:
%e A135877 [0,0,1,0,1,0,1];
%e A135877 take reverse partial sums to get row 2:
%e A135877 [3,3,3,2,2,1,1];
%e A135877 insert zeros at positions [0,1,3,5,8,11]:
%e A135877 [0,0,3,0,3,0,3,2,0,2,1,0,1];
%e A135877 take reverse partial sums to get row 3:
%e A135877 [15,15,15,12,12,9,9,6,4,4,2,1,1];
%e A135877 insert zeros at positions [0,1,3,5,8,11,15,19]:
%e A135877 [0,0,15,0,15,0,15,12,0,12,9,0,9,6,4,0,4,2,1,0,1];
%e A135877 take reverse partial sums to get row 4:
%e A135877 [105,105,105,90,90,75,75,60,48,48,36,27,27,18,12,8,8,4,2,1,1].
%o A135877 (PARI) {T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[0]; 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])} /* for(n=0,8,for(k=0,n*(n+1),print1(T(n,k),","));print("")) */
%Y A135877 Cf. A001147; A135876; variants: A135879, A127452, A125781.
%K A135877 nonn,tabl
%O A135877 0,5
%A A135877 _Paul D. Hanna_, Dec 14 2007