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 A127714 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. 5
 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, 58, 72, 72, 86, 86, 86, 1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645, 1, 6, 20, 48, 90, 132, 132, 196, 293, 434, 619, 804, 804 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Table of n, a(n) for n=0..68. FORMULA T(n,n) = A000108(n); A009766 (Catalan's triangle) forms lower left sub-triangle; T(n+1,2*n+1) = A127632(n), where g.f. of A127632 is: 1/(1+sqrt(2*sqrt(1-4x)-1). T(n,n*(n+1)/2) = A127716(n). T(n,(n+1)*(n+2)/2-1) = A127715(n). EXAMPLE Triangle begins: 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, 58, 72, 72, 86, 86, 86; 1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645; 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; ... Obtain row n from row n-1 by inserting zeros in row n-1 at positions: [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], and then take partial sums; illustrated by the following examples. Obtain row 3 from row 2 by inserting zeros at positions [3,6,8,9], and then take partial sums: [1, 2, 2, 0, 3, 3, 0, 3, 0, 0]; [1, 3, 5, 5, 8,11,11,14,14,14]; Obtain row 4 from row 3 by inserting zeros at positions [4,8,11,13,14], and then take partial sums: [1, 3, 5, _5, _0, _8, 11, 11, _0, 14, 14, _0, 14, _0, _0]; [1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86]. PROG (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("")) */} CROSSREFS Cf. A127715, A127716; A127632, A000108, A009766; variants: A127452, A127420. Sequence in context: A236465 A131619 A048485 * A283763 A357622 A357621 Adjacent sequences: A127711 A127712 A127713 * A127715 A127716 A127717 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jan 24 2007 STATUS approved

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Last modified June 7 00:08 EDT 2023. Contains 363151 sequences. (Running on oeis4.)