A125865 -id:A125865 - cp's OEIS frontend

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

Paul D. Hanna, Dec 13 2006

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

			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; ...

Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 16.

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

T[n_, k_] := T[n, k] = If[Or[n == 0, k == 0], 1, Sum[T[n - 1, j + k], {j, 0, k}]];
Table[T[#, k] &[n - k + 1], {n, 0, 9}, {k, 0, n + 1}] (* Michael De Vlieger, Dec 10 2024, after PARI *)

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

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.

A125859 Antidiagonal sums of table A125860.

Original entry on oeis.org

1, 2, 4, 10, 35, 184, 1531, 21080, 497017, 20533486, 1508839043, 199272672334, 47686000150774, 20817464210086523, 16678749474397158418, 24657143458135746104239, 67591557017940565183386368

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Cf. A125860; A016121, A125862, A125863, A125864, A125865, A125861.

A125861 Main diagonal of table A125860.

Original entry on oeis.org

1, 2, 12, 178, 5990, 435801, 66593821, 20997402098, 13512727916532, 17629371074833300, 46432767742317108086, 246240366959004185679198, 2624854986865673643625591411, 56179604057909797695704800461149

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Cf. A125860; A097712; columns: A016121, A125862, A125863, A125864, A125865; A125859 (antidiagonal sums).

A125862 Column 2 of table A125860; also equals row sums of matrix power A097712^2.

Original entry on oeis.org

1, 3, 12, 69, 612, 8853, 217041, 9245253, 695682342, 93580638024, 22713677612832, 10025101876435413, 8100572528598910191, 12054728928174188426943, 33214476295395054879355617, 170255688895444623691322464599

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^3, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums).

A125863 Column 3 of table A125860; also equals row sums of matrix power A097712^3.

Original entry on oeis.org

1, 4, 22, 178, 2251, 46663, 1640572, 100152049, 10794383587, 2079805452133, 723492192295786, 458149292979837523, 531871667833026397222, 1138955362720160687114704, 4523369812874327770490887837

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^4, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125862, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums).

A125864 Column 4 of table A125860; also equals row sums of matrix power A097712^4.

Original entry on oeis.org

1, 5, 35, 365, 5990, 161525, 7387640, 586285040, 82090572095, 20540291522675, 9278896006526795, 7632398133742637255, 11514756687812563119530, 32063466203746720003813970, 165699104606274900865952221145

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^5, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125862, A125863, A125865; A125861 (diagonal), A125859 (antidiagonal sums).

cp's OEIS Frontend

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

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A125859 Antidiagonal sums of table A125860.

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A125861 Main diagonal of table A125860.

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A125862 Column 2 of table A125860; also equals row sums of matrix power A097712^2.

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A125863 Column 3 of table A125860; also equals row sums of matrix power A097712^3.

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A125864 Column 4 of table A125860; also equals row sums of matrix power A097712^4.

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