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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

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

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.

EXAMPLE

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.

PROG

(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(""))

CROSSREFS

Cf. A129867, A130469; variant: A131338.

Sequence in context: A213234 A034807 A275111 * A135062 A088428 A025838

Adjacent sequences:  A182958 A182959 A182960 * A182962 A182963 A182964

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Dec 31 2010

STATUS

approved

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Last modified February 22 06:17 EST 2018. Contains 299430 sequences. (Running on oeis4.)