0,5

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(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.

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);

ex., 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; ...

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

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

Sequence in context: A105556 A078920 A117396 this_sequence A125800 A076241 A139347

Adjacent sequences: A125857 A125858 A125859 this_sequence A125861 A125862 A125863

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 13 2006