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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A109152 Triangle T, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) for n>k+1>=1, with T(n,n) = 1 and T(n+1,n) = n+1 for n>=0, where T^2 is the matrix square of T. 7
1, 1, 1, 2, 2, 1, 6, 4, 3, 1, 22, 14, 6, 4, 1, 94, 56, 24, 8, 5, 1, 450, 262, 102, 36, 10, 6, 1, 2366, 1348, 516, 160, 50, 12, 7, 1, 13450, 7574, 2814, 868, 230, 66, 14, 8, 1, 81802, 45616, 16752, 4968, 1330, 312, 84, 16, 9, 1, 527826, 292442, 106026, 31148, 7930, 1914, 406, 104, 18, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..350

FORMULA

T^(m+1) = SHIFT_UP(T^m - T^(m-1)) - D*T^(m-1) for all m where diagonal matrix D = [0, 1, 2, 3, ...] and SHIFT_UP shifts each column up 1 row.

Recurrence:

T(n,k) = Sum_{j=k..n-1} T(n-1, j) * T(j, k) for k=0..n-2 for n>1 with T(n+1,n) = n+1 and T(n,n) = 1 for n>=0. - Paul D. Hanna, Jan 13 2016

EXAMPLE

Triangle T begins:

1;

1, 1;

2, 2, 1;

6, 4, 3, 1;

22, 14, 6, 4, 1;

94, 56, 24, 8, 5, 1;

450, 262, 102, 36, 10, 6, 1;

2366, 1348, 516, 160, 50, 12, 7, 1;

13450, 7574, 2814, 868, 230, 66, 14, 8, 1;

81802, 45616, 16752, 4968, 1330, 312, 84, 16, 9, 1;

527826, 292442, 106026, 31148, 7930, 1914, 406, 104, 18, 10, 1; ...

Matrix square T^2 starts:

      1;

      2       1;

      6       4      1;

     22      14      6      1;

     94      56     24      8     1;

    450     262    102     36    10     1;

   2366    1348    516    160    50    12    1;

  13450    7574   2814    868   230    66   14   1;

  81802   45616  16752   4968  1330   312   84  16   1;

527826  292442 106026  31148  7930  1914  406 104  18  1;

3590294 1978788 712260 205888 52130 11820 2632 512 126 20 1; ...

which equals SHIFT_UP(T) - D where

D is the diagonal matrix [0,1,2,3,...].

PROG

(PARI) {T(n, k) = local(M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1)))); if(n<k||k<0, 0, if(n==k||n==0, 1, if(n==k+1, n, (M^2)[n, k+1])))}

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

(PARI) {T(n, k) = if(n==k, 1, if(n==k+1, n, sum(j=k, n-1, T(n-1, j) * T(j, k) )))}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 13 2016

CROSSREFS

Cf. A109153 (column 0), A109154 (column 1), A109155 (column 2), A109156 (row sums).

Cf. A152391.

Sequence in context: A178886 A065052 A128308 * A130469 A106381 A064784

Adjacent sequences:  A109149 A109150 A109151 * A109153 A109154 A109155

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jun 20 2005

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 August 7 12:19 EDT 2020. Contains 336276 sequences. (Running on oeis4.)