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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.
5
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
0,5
COMMENTS
Last term in each row forms A107877, the number of subpartitions of the partition consisting of the triangular numbers.
LINKS
EXAMPLE
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:
[1, 1, 2, 7, 37, 268, 2496, 28612, 391189, 6230646, 113521387, ...]
which satisfies the g.f.:
1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^3 + 7*x^3*(1-x)^6 +
37*x^4*(1-x)^10 + 268*x^5*(1-x)^15 + 2496*x^6*(1-x)^21 +...
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 *)
PROG
(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))))
CROSSREFS
Cf. A107877 (leading edge); diagonals: A127497, A127498.
Sequence in context: A026268 A089258 A004065 * A376626 A350189 A289778
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 16 2007
STATUS
approved