A127496 Triangle, read by rows of n*(n+1)/2 + 1 terms, generated by the following rule: start with a single '1' in row n=0; subsequently, row n+1 equals the partial sums of row n with the final term repeated n+1 more times at the end.
1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 7, 7, 7, 7, 1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37, 1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268, 1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496
Offset: 0
Examples
To obtain row 4 from row 3: [1, 3, _5, _7, _7, _7, __7]; take partial sums with final term '37' repeated 4 more times: [1, 4, _9, 16, 23, 30, _37, _37, _37, _37, _37]. To obtain row 5, take partial sums of row 4 with the final term '268' repeated 5 more times at the end: [1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268,268,268,268,268]. Triangle begins: 1; 1, 1; 1, 2, 2, 2; 1, 3, 5, 7, 7, 7, 7; 1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37; 1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268; 1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496; Final term in rows forms A107877 which satisfies the g.f. 1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^3 + ...
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
nxt[h_] :=Module[{c = Accumulate[h]}, Join[c, PadRight[{}, c[[2]], c[[-1]]]]]; Join[{1},Flatten[NestList[nxt,{1,1},5]]] (* Harvey P. Dale, Mar 10 2020 *) -
PARI
T(n,k)=if(n<0 || k<0 || k>n*(n+1)/2,0,if(k==0,1, if(k<=n*(n-1)/2,T(n,k-1)+T(n-1,k),T(n,k-1))))
Comments