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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.

%I A127496 #20 May 09 2025 11:39:07
%S A127496 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 A127496 53,83,120,157,194,231,268,268,268,268,268,268,1,6,20,50,103,186,306,
%U A127496 463,657,888,1156,1424,1692,1960,2228,2496,2496,2496,2496,2496,2496,2496
%N 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.
%C A127496 Last term in each row forms A107877, the number of subpartitions of the partition consisting of the triangular numbers.
%H A127496 Harvey P. Dale, <a href="/A127496/b127496.txt">Table of n, a(n) for n = 0..1000</a>
%e A127496 To obtain row 4 from row 3:
%e A127496   [1, 3, _5, _7, _7, _7, __7];
%e A127496 take partial sums with final term '37' repeated 4 more times:
%e A127496   [1, 4, _9, 16, 23, 30, _37, _37, _37, _37, _37].
%e A127496 To obtain row 5, take partial sums of row 4 with the final term '268' repeated 5 more times at the end:
%e A127496   [1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268,268,268,268,268].
%e A127496 Triangle begins:
%e A127496   1;
%e A127496   1, 1;
%e A127496   1, 2, 2, 2;
%e A127496   1, 3, 5, 7, 7, 7, 7;
%e A127496   1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37;
%e A127496   1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268;
%e A127496   1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496;
%e A127496 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 + ...
%t A127496 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 A127496 (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 A127496 Cf. A107877 (leading edge); diagonals: A127497, A127498.
%Y A127496 Cf. A305605, A305601.
%K A127496 nonn,tabf
%O A127496 0,5
%A A127496 _Paul D. Hanna_, Jan 16 2007