A125860 Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 12, 4, 1, 1, 86, 69, 22, 5, 1, 1, 698, 612, 178, 35, 6, 1, 1, 9551, 8853, 2251, 365, 51, 7, 1, 1, 226592, 217041, 46663, 5990, 651, 70, 8, 1, 1, 9471845, 9245253, 1640572, 161525, 13131, 1057, 92, 9, 1, 1, 705154187

Offset

0,5

Comments

Triangle A097712 satisfies: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1) for n > 0, k > 0, with A097712(n,0)=A097712(n,n)=1 for n >= 0. Column 1 equals A016121, which counts the sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

T(2, n) = (n+1)*A005408(n) - Sum_{i=0..n} A001477(i) = (n+1)*(2*n+1) - A000217(n) = (n+1)*(3*n+2)/2; T(3, n) = (n+1)*A001106(n+1) - Sum_{i=0..n} A001477(i) = (n+1)*((n+1)*(7*n+2)/2) - A000217(n) = (n+1)*(7*n^2 + 8*n + 2)/2. - Bruno Berselli, Apr 25 2010

Links

Table of n, a(n) for n=0..56.

Formula

T(n,k) = Sum_{j=0..k} T(n-1, j+k) for n > 0, with T(0,n)=T(n,0)=1 for n >= 0.

Example

Recurrence is illustrated by:
T(4,1) = T(3,1) + T(3,2) = 17 + 69 = 86;
T(4,2) = T(3,2) + T(3,3) + T(3,4) = 69 + 178 + 365 = 612;
T(4,3) = T(3,3) + T(3,4) + T(3,5) + T(3,6) = 178 + 365 + 651 + 1057 = 2251.
Rows of this table begin:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,...;
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, ...;
1, 17, 69, 178, 365, 651, 1057, 1604, 2313, 3205, 4301, 5622, 7189,..;
1, 86, 612, 2251, 5990, 13131, 25291, 44402, 72711, 112780, 167486,..;
1, 698, 8853, 46663, 161525, 435801, 996583, 2025458, 3768273, ...;
1, 9551, 217041, 1640572, 7387640, 24530016, 66593821, 156664796, ...;
1, 226592, 9245253, 100152049, 586285040, 2394413286, 7713533212, ...;
1, 9471845, 695682342, 10794383587, 82090572095, 412135908606, ...;
1, 705154187, 93580638024, 2079805452133, 20540291522675, ...;
1, 94285792211, 22713677612832, 723492192295786, 9278896006526795,...;
1, 22807963405043, 10025101876435413, 458149292979837523, ...;
...
where column k equals the row sums of matrix power A097712^k for k >= 0.
Triangle A097712 begins:
  1;
  1,      1;
  1,      3,       1;
  1,      8,       7,       1;
  1,     25,      44,      15,       1;
  1,    111,     346,     208,      31,      1;
  1,    809,    4045,    3720,     912,     63,     1;
  1,  10360,   77351,   99776,   35136,   3840,   127,   1;
  1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255; ...
where A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1);
e.g., A097712(5,2) = A097712(4,2) + [A097712^2](4,1) = 44 + 302 = 346.
Matrix square A097712^2 begins:
     1;
     2,     1;
     5,     6,     1;
    17,    37,    14,     1;
    86,   302,   193,    30,    1;
   698,  3699,  3512,   881,   62,   1;
  9551, 73306, 96056, 34224, 3777, 126, 1; ...
Matrix cube A097712^3 begins:
       1;
       3,      1;
      12,      9,      1;
      69,     87,     21,      1;
     612,   1146,    447,     45,    1;
    8853,  22944,  12753,   2019,   93,   1;
  217041, 744486, 549453, 120807, 8595, 189, 1; ...

Prog

(PARI) T(n, k)=if(n==0 || k==0, 1, sum(j=0, k, T(n-1, j+k)))

Crossrefs

Cf. A097712; columns: A016121, A125862, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums). Variants: A125790, A125800.

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) and similar: A081436, A005920, A005945, A006003. - Bruno Berselli, Apr 25 2010

Sequence in context: A241579 A308292 A117396 * A294585 A283674 A294758

Adjacent sequences: A125857 A125858 A125859 * A125861 A125862 A125863

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 13 2006

Status

approved