A127496 - OEIS

%I #13 Mar 10 2020 17:31:11

%S 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,

%T 53,83,120,157,194,231,268,268,268,268,268,268,1,6,20,50,103,186,306,

%U 463,657,888,1156,1424,1692,1960,2228,2496,2496,2496,2496,2496,2496,2496

%N 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.

%C Last term in each row forms A107877, the number of subpartitions of the partition consisting of the triangular numbers.

%H Harvey P. Dale, <a href="/A127496/b127496.txt">Table of n, a(n) for n = 0..1000</a>

%e To obtain row 4 from row 3:

%e [1, 3, _5, _7, _7, _7, __7];

%e take partial sums with final term '37' repeated 4 more times:

%e [1, 4, _9, 16, 23, 30, _37, _37, _37, _37, _37].

%e To obtain row 5, take partial sums of row 4 with the final term '268'

%e repeated 5 more times at the end:

%e [1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268,268,268,268,268].

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 2, 2;

%e 1, 3, 5, 7, 7, 7, 7;

%e 1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37;

%e 1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268;

%e 1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496;

%e Final term in rows forms A107877:

%e [1, 1, 2, 7, 37, 268, 2496, 28612, 391189, 6230646, 113521387, ...]

%e which satisfies the g.f.:

%e 1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^3 + 7*x^3*(1-x)^6 +

%e 37*x^4*(1-x)^10 + 268*x^5*(1-x)^15 + 2496*x^6*(1-x)^21 +...

%t 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 *)

%o (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))))

%Y Cf. A107877 (leading edge); diagonals: A127497, A127498.

%Y Cf. A305605, A305601.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jan 16 2007