login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A101479 Triangular matrix T, read by rows, where row n equals row (n-1) of T^(n-1) after appending '1' for the main diagonal. 25
1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 19, 9, 3, 1, 1, 191, 70, 18, 4, 1, 1, 2646, 795, 170, 30, 5, 1, 1, 46737, 11961, 2220, 335, 45, 6, 1, 1, 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1, 25330125, 5051866, 758814, 92652, 9730, 924, 84, 8, 1, 1, 735180292, 132523155 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Remarkably, T equals the product of these triangular matrices: T = A107867*A107862^-1 = A107870*A107867^-1 = A107873*A107870^-1; reversing the order of these products yields triangle A107876.

LINKS

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

EXAMPLE

Triangle begins:

1;

1, 1;

1, 1, 1;

3, 2, 1, 1;

19, 9, 3, 1, 1;

191, 70, 18, 4, 1, 1;

2646, 795, 170, 30, 5, 1, 1;

46737, 11961, 2220, 335, 45, 6, 1, 1;

1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1;

25330125, 5051866, 758814, 92652, 9730, 924, 84, 8, 1, 1;

735180292, 132523155, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1; ...

Row 4 starts with row 3 of T^3 which begins:

1;

3, 1;

6, 3, 1;

19, 9, 3, 1; ...

row 5 starts with row 4 of T^4 which begins:

1;

4, 1;

10, 4, 1;

34, 14, 4, 1;

191, 70, 18, 4, 1; ...

An ALTERNATE GENERATING METHOD is illustrated as follows.

For row 4:

Start with a '1' and append 2 zeros,

take partial sums and append 1 zero,

take partial sums thrice more, resulting in:

1, 0, 0;

1, 1, 1, 0;

1, 2, 3, 3;

1, 3, 6, 9;

1, 4,10,19.

Final nonzero terms form row 4: [19,9,3,1,1].

For row 5:

Start with a '1' and append 3 zeros,

take partial sums and append 2 zeros,

take partial sums and append 1 zero,

take partial sums thrice more, resulting in:

1, 0, 0, 0;

1, 1, 1, 1, 0,  0;

1, 2, 3, 4, 4,  4,  0;

1, 3, 6,10,14, 18, 18;

1, 4,10,20,34, 52, 70;

1, 5,15,35,69,121,191;

where the final nonzero terms form row 5: [191,70,18,4,1,1].

Likewise, for row 6:

1, 0, 0, 0,  0;

1, 1, 1, 1,  1,  0,  0,  0;

1, 2, 3, 4,  5,  5,  5,  5,   0,   0;

1, 3, 6,10, 15, 20, 25, 30,  30,  30,   0;

1, 4,10,20, 35, 55, 80,110, 140, 170, 170;

1, 5,15,35, 70,125,205,315, 455, 625, 795;

1, 6,21,56,126,251,456,771,1226,1851,2646;

where the final nonzero terms form row 6: [2646,795,170,30,5,1,1].

Continuing in this way generates all rows of this triangle.

PROG

(PARI) {T(n, k) = my(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j] = (A^(i-2))[i-1, j]); )); A=B); return(A[n+1, k+1])}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(PARI) {T(n, k) = my(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)*(n-2)/2-(n-j-1)*(n-j-2)/2; A = Vec((Polrev(A)+x*O(x^p))/(1-x))); A = Vec((Polrev(A) +x*O(x^p)) / (1-x) ); A[p+1]}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Columns are A101481, A101482, A101483, row sums form A101484.

Cf. A107876 (dual triangle).

Cf. A304184, A304185, A304186, A304187.

Sequence in context: A214742 A204124 A316674 * A136170 A245188 A137241

Adjacent sequences:  A101476 A101477 A101478 * A101480 A101481 A101482

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 21 2005, Jul 26 2006, May 27 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 23 16:45 EST 2020. Contains 331172 sequences. (Running on oeis4.)