A125800 -id:A125800 - cp's OEIS frontend

A125801 Column 3 of table A125800; also equals row sums of matrix power A078122^3.

1, 4, 22, 238, 5827, 342382, 50110483, 18757984045, 18318289003447, 47398244089264546, 329030840161393127680, 6190927493941741957366099, 318447442589056401640929570895, 45106654667152833836835578059359838

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Number of partitions of 3^n into powers of 3, excluding the trivial partition 3^n=3^n. - Valentin Bakoev, Feb 20 2009

			To obtain t_3(5,1) we use the table T, defined as T(i,j) = t_3(i,j), for i=1,2,...,5(=n), and j=0,1,2,...,81(= k*m^{n-1}). It is 1,1,1,1,1,1,...1; 1,4,7,10,13,...,82; 1,22,70,145,247,376,532,715,925,1162; 1,238,1393,4195; 1,5827; Column 1 contains the first 5 terms of A125801. - _Valentin Bakoev_, Feb 20 2009

Alois P. Heinz, Table of n, a(n) for n = 0..40
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

Cf. A125800, A078122; other columns: A078125, A078124, A125802, A125803.

g:= proc(b, n, k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add (g(b-t, n, k) *binomial (n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1, n+1,3)-1: seq(a(n), n=0..25); # Alois P. Heinz, Feb 27 2009

T[0, ] = T[, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k];
a[n_] := T[n, 3]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* Jean-François Alcover, Jan 21 2017 *)

a(n)=local(p=3,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

Denote the sum: m^n  +m^n + ... + m^n, k times, by k*m^n (m > 1, n > 0 and k are positive integers). The general formula for the number of all partitions of the sum k*m^n into powers of m smaller than m^n, is t_m(n, k)= 1 when n=1 or k=0, or = t_m(n, k-1) + Sum_{j=1..m} t_m(n-1, (k-1)*n+j)}, when n > 1 and k > 0. A125801 is obtained for m=3 and n=1,2,3,... - Valentin Bakoev, Feb 20 2009
From Valentin Bakoev, Feb 20 2009: (Start)
Adding 1 to the terms of A125801 we obtain A078125.
For given m, the general formula for t_m(n, k) and the corresponding tables T, computed as in the example, determine a family of related sequences (placed in the rows or in the columns of T). For example, the sequences from the 3rd, 4th, etc. rows of the given table are not represented in the OEIS till now. (End)
a(n) = A145515(n+1,3)-1. - Alois P. Heinz, Feb 27 2009

A125802 Column 4 of table A125800; also equals row sums of matrix power A078122^4.

Original entry on oeis.org

1, 5, 35, 485, 15200, 1144664, 215155493, 103674882878, 130648799730635, 437302448840089232, 3936208033244539574405, 96244898501021613327012635, 6446494058446469307795159512465, 1191218783863555524342034469450207222

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Cf. A125800, A078122; other columns: A078125, A078124, A125801, A125803.

a(n)=local(p=4,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125803 Column 5 of table A125800; also equals row sums of matrix power A078122^5.

Original entry on oeis.org

1, 6, 51, 861, 32856, 3013980, 690729981, 406279238154, 625750288074015, 2563196032703643450, 28270494794022487841733, 848050124165724284639262951, 69769378541879435090796205851249

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Cf. A125800, A078122; other columns: A078125, A078124, A125801, A125802.

a(n)=local(p=5,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125804 Main diagonal of table A125800.

Original entry on oeis.org

1, 2, 12, 238, 15200, 3013980, 1828979530, 3373190565626, 18837339867421686, 317817051628161116674, 16176220447967300610844988, 2481251352301850541661479580329, 1146112129196402690505198891390847384

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Table A125800 is related to partitions into powers of 3; column k of A125800 equals row sums of matrix power A078122^k, where triangle A078122 shifts left one column under matrix cube.

Cf. A125800, A078122; columns: A078125, A078124, A125801, A125802, A125803; A125805 (antidiagonal sums).

a(n)=local(q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^n)[n+1,c+1]))

A125805 Antidiagonal sums of table A125800.

Original entry on oeis.org

1, 2, 4, 10, 41, 361, 7741, 417212, 57581062, 20688363559, 19625079296963, 49742424992663959, 340292157995636104240, 6337196928437059669994069, 323627960380394115802942263514, 45610724032832026072070666274435391

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Table A125800 is related to partitions into powers of 3; column k of A125800 equals row sums of matrix power A078122^k, where triangle A078122 shifts left one column under matrix cube.

Cf. A125800, A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal).

a(n)=local(q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^(c+1))[n-c+1,1]))

A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 36, 35, 16, 5, 1, 1, 202, 201, 84, 25, 6, 1, 1, 1828, 1827, 656, 165, 36, 7, 1, 1, 27338, 27337, 8148, 1625, 286, 49, 8, 1, 1, 692004, 692003, 167568, 25509, 3396, 455, 64, 9, 1, 1, 30251722, 30251721, 5866452, 664665, 64350, 6321, 680, 81, 10, 1

Offset: 0

Paul D. Hanna, Dec 10 2006, corrected Dec 12 2006

Determinant of n X n upper left submatrix is 2^[n(n-1)(n-2)/6] (see A125791). Related to partitions of numbers into powers of 2 (see A078121). Triangle A078121 shifts left one column under matrix square.

			Recurrence T(n,k) = T(n,k-1) + T(n-1,2*k) is illustrated by:
  T(4,3) = T(4,2) + T(3,6) = 201 + 455 = 656;
  T(5,3) = T(5,2) + T(4,6) = 1827 + 6321 = 8148;
  T(6,3) = T(6,2) + T(5,6) = 27337 + 140231 = 167568.
Rows of this table begin:
  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, ...;
  1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, ...;
  1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, ...;
  1, 36, 201, 656, 1625, 3396, 6321, 10816, 17361, 26500, 38841, ...;
  1, 202, 1827, 8148, 25509, 64350, 140231, 274856, 497097, ...;
  1, 1828, 27337, 167568, 664665, 2026564, 5174449, 11622976, ...;
  1, 27338, 692003, 5866452, 29559717, 109082974, 326603719, ...;
  1, 692004, 30251721, 356855440, 2290267225, 10243585092, ...; ...
Triangle A078121 begins:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,  10,  16,   8,   1;
  1,  36,  84,  64,  16,  1;
  1, 202, 656, 680, 256, 32, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078121 equals column 2^k-1 of this table A125790.
Matrix cube A078121^3 begins:
     1;
     3,    1;
     9,    6,    1;
    35,   36,   12,   1;
   201,  286,  144,  24,  1;
  1827, 3396, 2300, 576, 48, 1; ...
where row sums form column 3 of this table A125790,
and column 0 of A078121^3 forms column 2 of this table A125790.

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 5.

Cf. A078121; A002577; A125791; columns: A002577, A125792, A125793, A125794, A125795, A125796; diagonals: A125797, A125798; A125799 (antidiagonal sums); related table: A125800 (q=3).

T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, ] = T[, 0] = 1; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 15 2015 *)

{T(n,k,p=0,q=2)=local(A=Mat(1), B); if(n

T(n,k) = T(n,k-1) + T(n-1,2*k) for n>0, k>0, with T(0,n)=T(n,0)=1 for n>=0.

Conjecture: g.f. for n-th row is (Sum_{i=0..n-1} x^i Sum_{j=0..i} binomial(n+1,j)*T(n,i-j)*(-1)^j)/(1-x)^(n+1) for n > 0. - Mikhail Kurkov, May 03 2025

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

Original entry on oeis.org

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.

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A125801 Column 3 of table A125800; also equals row sums of matrix power A078122^3.

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A125802 Column 4 of table A125800; also equals row sums of matrix power A078122^4.

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A125803 Column 5 of table A125800; also equals row sums of matrix power A078122^5.

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A125804 Main diagonal of table A125800.

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A125805 Antidiagonal sums of table A125800.

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A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

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

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