This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A125860 #20 Dec 10 2024 13:27:57
%S A125860 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,
%T A125860 1,9551,8853,2251,365,51,7,1,1,226592,217041,46663,5990,651,70,8,1,1,
%U A125860 9471845,9245253,1640572,161525,13131,1057,92,9,1,1,705154187
%N A125860 Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.
%C A125860 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.
%C A125860 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
%H A125860 Olivier Danvy, <a href="https://arxiv.org/abs/2412.03127">Summa Summarum: Moessner's Theorem without Dynamic Programming</a>, arXiv:2412.03127 [cs.DM], 2024. See p. 16.
%F A125860 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.
%e A125860 Recurrence is illustrated by:
%e A125860 T(4,1) = T(3,1) + T(3,2) = 17 + 69 = 86;
%e A125860 T(4,2) = T(3,2) + T(3,3) + T(3,4) = 69 + 178 + 365 = 612;
%e A125860 T(4,3) = T(3,3) + T(3,4) + T(3,5) + T(3,6) = 178 + 365 + 651 + 1057 = 2251.
%e A125860 Rows of this table begin:
%e A125860 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e A125860 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,...;
%e A125860 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, ...;
%e A125860 1, 17, 69, 178, 365, 651, 1057, 1604, 2313, 3205, 4301, 5622, 7189,..;
%e A125860 1, 86, 612, 2251, 5990, 13131, 25291, 44402, 72711, 112780, 167486,..;
%e A125860 1, 698, 8853, 46663, 161525, 435801, 996583, 2025458, 3768273, ...;
%e A125860 1, 9551, 217041, 1640572, 7387640, 24530016, 66593821, 156664796, ...;
%e A125860 1, 226592, 9245253, 100152049, 586285040, 2394413286, 7713533212, ...;
%e A125860 1, 9471845, 695682342, 10794383587, 82090572095, 412135908606, ...;
%e A125860 1, 705154187, 93580638024, 2079805452133, 20540291522675, ...;
%e A125860 1, 94285792211, 22713677612832, 723492192295786, 9278896006526795,...;
%e A125860 1, 22807963405043, 10025101876435413, 458149292979837523, ...;
%e A125860 ...
%e A125860 where column k equals the row sums of matrix power A097712^k for k >= 0.
%e A125860 Triangle A097712 begins:
%e A125860 1;
%e A125860 1, 1;
%e A125860 1, 3, 1;
%e A125860 1, 8, 7, 1;
%e A125860 1, 25, 44, 15, 1;
%e A125860 1, 111, 346, 208, 31, 1;
%e A125860 1, 809, 4045, 3720, 912, 63, 1;
%e A125860 1, 10360, 77351, 99776, 35136, 3840, 127, 1;
%e A125860 1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255; ...
%e A125860 where A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1);
%e A125860 e.g., A097712(5,2) = A097712(4,2) + [A097712^2](4,1) = 44 + 302 = 346.
%e A125860 Matrix square A097712^2 begins:
%e A125860 1;
%e A125860 2, 1;
%e A125860 5, 6, 1;
%e A125860 17, 37, 14, 1;
%e A125860 86, 302, 193, 30, 1;
%e A125860 698, 3699, 3512, 881, 62, 1;
%e A125860 9551, 73306, 96056, 34224, 3777, 126, 1; ...
%e A125860 Matrix cube A097712^3 begins:
%e A125860 1;
%e A125860 3, 1;
%e A125860 12, 9, 1;
%e A125860 69, 87, 21, 1;
%e A125860 612, 1146, 447, 45, 1;
%e A125860 8853, 22944, 12753, 2019, 93, 1;
%e A125860 217041, 744486, 549453, 120807, 8595, 189, 1; ...
%t A125860 T[n_, k_] := T[n, k] = If[Or[n == 0, k == 0], 1, Sum[T[n - 1, j + k], {j, 0, k}]];
%t A125860 Table[T[#, k] &[n - k + 1], {n, 0, 9}, {k, 0, n + 1}] (* _Michael De Vlieger_, Dec 10 2024, after PARI *)
%o A125860 (PARI) T(n,k)=if(n==0 || k==0,1,sum(j=0,k,T(n-1,j+k)))
%Y A125860 Cf. A097712; columns: A016121, A125862, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums). Variants: A125790, A125800.
%Y A125860 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
%K A125860 nonn,tabl
%O A125860 0,5
%A A125860 _Paul D. Hanna_, Dec 13 2006