A182961 Triangle, read by rows, where terms in row n equal the partial sums of row n-1 with 1's inserted at positions [0,n,2n-1,3n-3,4n-6,5n-10,...,n(n+1)/2-1] for n>0, with T(0,0)=1.

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 5, 1, 1, 2, 4, 1, 5, 8, 1, 9, 1, 14, 1, 1, 2, 4, 8, 1, 9, 14, 22, 1, 23, 32, 1, 33, 1, 47, 1, 1, 2, 4, 8, 16, 1, 17, 26, 40, 62, 1, 63, 86, 118, 1, 119, 152, 1, 153, 1, 200, 1

Offset: 0

Author

Paul D. Hanna, Dec 31 2010

Keywords

• nonn
• tabf

Examples

			This triangle T(n,k), where k=0..n(n+1)/2 in row n>=0, begins:
1;
(1),1;
(1),1,(1),2;
(1),1,2,(1),3,(1),5;
(1),1,2,4,(1),5,8,(1),9,(1),14;
(1),1,2,4,8,(1),9,14,22,(1),23,32,(1),33,(1),47;
(1),1,2,4,8,16,(1),17,26,40,62,(1),63,86,118,(1),119,152,(1),153,(1),200;
(1),1,2,4,8,16,32,(1),33,50,76,116,178,(1),179,242,328,446,(1),447,566,718,(1),719,872,(1),873,(1),1073;
...
where row n is equal to the partial sums of terms in row n-1, with 1's inserted at positions [0,n,2n-1,3n-3,4n-6,5n-10,...,n(n+1)/2-1].
The row sums and rightmost border form sequence A129867, which equals the row sums of triangle A130469.
Triangle A130469 begins:
1;
1, 1;
2, 2, 1;
6, 4, 3, 1;
24, 12, 6, 4, 1;
120, 48, 18, 8, 5, 1;
720, 240, 72, 24, 10, 6, 1; ...
which has the same row sums as this triangle.
		

Links

• Paul D. Hanna, Rows n = 0..30, flattened.

Crossrefs

Cf. A129867, A130469; variant: A131338.

Programs

PARI

{T(n,k)=local(A=[1],B); for(m=0,n, t=0;B=[];
for(j=0,#A-1, if(j==t*m-t*(t+1)/2, t+=1;B=concat(B,1)); B=concat(B,A[j+1]));
A=Vec( Ser(B)/(1-x+O(x^#B)) ) ); if(k+1>#A, 0, B[k+1])}
for(n=0,12,for(k=0,n*(n+1)/2,print1(T(n, k), ", ")); print(""))

Formula

Row sums equal A129867;

n-th row sum = 1 + Sum_{k=1..n} k*(n-k+1)!.

T(n,n(n+1)/2) = A129867(n) for n>0, with T(0,0) = 1.