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%I #8 Jun 14 2017 00:24:19
%S 1,1,1,1,1,2,2,3,3,3,1,3,5,5,8,11,11,14,14,14,1,4,9,14,14,22,33,44,44,
%T 58,72,72,86,86,86,1,5,14,28,42,42,64,97,141,185,185,243,315,387,387,
%U 473,559,559,645,645,645,1,6,20,48,90,132,132,196,293,434,619,804,804
%N Triangle, read by rows of (n+1)*(n+2)/2 terms, generated by the following rule. Start with a single '1' in row n=0; from then on, obtain row n from row n-1 by inserting zeros in row n-1 at positions: {(j+1)*n - j*(j-1)/2 | j=0..n} and then take partial sums.
%F T(n,n) = A000108(n);
%F A009766 (Catalan's triangle) forms lower left sub-triangle;
%F T(n+1,2*n+1) = A127632(n), where g.f. of A127632 is: 1/(1+sqrt(2*sqrt(1-4x)-1).
%F T(n,n*(n+1)/2) = A127716(n).
%F T(n,(n+1)*(n+2)/2-1) = A127715(n).
%e Triangle begins:
%e 1;
%e 1, 1, 1;
%e 1, 2, 2, 3, 3, 3;
%e 1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
%e 1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;
%e 1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645;
%e 1, 6, 20, 48, 90, 132, 132, 196, 293, 434, 619, 804, 804, 1047, 1362, 1749, 2136, 2136, 2609, 3168, 3727, 3727, 4372, 5017, 5017, 5662, 5662, 5662;
%e ...
%e Obtain row n from row n-1 by inserting zeros in row n-1 at positions:
%e [n,2*n,3*n-1,4*n-3,5*n-6,6*n-10,...,(j+1)*n - j*(j-1)/2,... | j=0..n],
%e and then take partial sums; illustrated by the following examples.
%e Obtain row 3 from row 2 by inserting zeros at positions [3,6,8,9],
%e and then take partial sums:
%e [1, 2, 2, 0, 3, 3, 0, 3, 0, 0];
%e [1, 3, 5, 5, 8,11,11,14,14,14];
%e Obtain row 4 from row 3 by inserting zeros at positions [4,8,11,13,14],
%e and then take partial sums:
%e [1, 3, 5, _5, _0, _8, 11, 11, _0, 14, 14, _0, 14, _0, _0];
%e [1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86].
%o (PARI) {T(n,k)=local(t);if(n<0 || k<0 || k>(n+1)*(n+2)/2-1,0, t=(sqrtint((2*n+3)^2-8*(k+1))-1)\2; if(k==0,1,if(issquare((2*n+3)^2-8*(k+1)),T(n,k-1),T(n,k-1)+T(n-1,k-n+t))))} {/* for(n=0,8,for(k=0,(n+1)*(n+2)/2-1,print1(T(n,k),","));print("")) */}
%Y Cf. A127715, A127716; A127632, A000108, A009766; variants: A127452, A127420.
%K nonn,tabl
%O 0,6
%A _Paul D. Hanna_, Jan 24 2007
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