A078122 -id:A078122 - cp's OEIS frontend

A078124 Second column, M(n+1,1) for n>=0, of infinite lower triangular matrix M defined in A078122.

1, 3, 12, 93, 1632, 68457, 7112055, 1879090014, 1287814075131, 2325758241901161, 11213788533232011006, 145939965725683888932081, 5174322925070232320838406581, 503750821963423009552527526376232

Offset: 0

Paul D. Hanna, Nov 18 2002

nonn

			a(1)=3 since the coefficient of x^6 in 1/Product_{j=0..inf}(1-x^(3^j)) = 1 + x + x^2 + 2x^3 + 2x^4 + 2x^5 + 3x^6 + ... is 3.

Alois P. Heinz, Table of n, a(n) for n = 0..40

Cf. A078121, A078122 (matrix shift when cubed), A078123, A078125.

m[i_, j_] := m[i, j]=If[j==0||i==j, 1, m3[i-1, j-1]]; m2[i_, j_] := m2[i, j]=Sum[m[i, k]m[k, j], {k, j, i}]; m3[i_, j_] := m3[i, j]=Sum[m[i, k]m2[k, j], {k, j, i}]; a[n_] := m[n+1, 1]

The partitions of 2*3^n into powers of 3, or, the coefficient of x^(2*3^n) in 1/Product_{j=0..inf}(1-x^(3^j)) (conjecture).

A111815 Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

Original entry on oeis.org

0, 1, 0, -1, 3, 0, -3, -3, 9, 0, 150, -9, -9, 27, 0, 1236, 450, -27, -27, 81, 0, -2555748, 3708, 1350, -81, -81, 243, 0, -64342116, -7667244, 11124, 4050, -243, -243, 729, 0, 5885700899760, -193026348, -23001732, 33372, 12150, -729, -729, 2187, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 3^k multiplied by column 0 (A111816) when ignoring zeros above the diagonal.

			Matrix log of A078122, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, 3/1!, 0;
-3/3!, -3/2!, 9/1!, 0;
150/4!, -9/3!, -9/2!, 27/1!, 0;
1236/5!, 450/4!, -27/3!, -27/2!, 81/1!, 0;
-2555748/6!, 3708/5!, 1350/4!, -81/3!, -81/2!, 243/1!, 0; ...

Cf. A078122, A111816 (column 0), A111840 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

PARI
```
T(n,k,q=3)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 3^k*T(n-k, 0) = A111816(n-k) for n>=k>=0.

A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, -1, -3, 150, 1236, -2555748, -64342116, 5885700899760, 442646611978752, -1737387344860364226240, -367706581563500487774720, 60788555325888838346137808787840, 34626906551623392401873575206240000, -237458311254822429335982538087618909465992960

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=3; the g.f. of column k of A078122^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

			E.g.f.: A(x) = x - 1/2!*x^2 - 3/3!*x^3 + 150/4!*x^4 + 1236/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(3*x)/2! + A(x)*A(3*x)*A(3^2*x)/3! +
A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...
Let G(x) be the g.f. of A078124 (column 1 of A078122), then
G(x) = 1 + 3*A(x) + 3^2*A(x)*A(3*x)/2! +
3^3*A(x)*A(3*x)*A(3^2*x)/3! +
3^4*A(x)*A(3*x)*A(3^2*x)*A(3^3*x)/4! + ...

Cf. A078122 (triangle), A078124, A111815 (matrix log); A110505 (q=-1), A111814 (q=2), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=3)=local(A=x/(1-x+x*O(x^n)));for(i=1,n, A=x/(1-x)/(1+sum(j=1,n,prod(k=1,j,subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(3^j*x)/(j+1).

A125800 Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 23, 12, 4, 1, 1, 239, 93, 22, 5, 1, 1, 5828, 1632, 238, 35, 6, 1, 1, 342383, 68457, 5827, 485, 51, 7, 1, 1, 50110484, 7112055, 342382, 15200, 861, 70, 8, 1, 1, 18757984046, 1879090014, 50110483, 1144664, 32856, 1393, 92, 9, 1

Offset: 0

Paul D. Hanna, Dec 10 2006

Determinant of n X n upper left submatrix is 3^(n*(n-1)*(n-2)/6).
This table is related to partitions of numbers into powers of 3 (see A078122).
Triangle A078122 shifts left one column under matrix cube.
Column 1 is A078125, which equals row sums of A078122;
column 2 is A078124, which equals row sums of A078122^2.

			Recurrence T(n,k) = T(n,k-1) + T(n-1,3*k) is illustrated by:
  T(3,3) = T(3,2) + T(2,9) = 93 + 145 = 238;
  T(4,3) = T(4,2) + T(3,9) = 1632 + 4195 = 5827;
  T(5,3) = T(5,2) + T(4,9) = 68457 + 273925 = 342382.
Rows of this table begin:
  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, ...;
  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ...;
  1, 23, 93, 238, 485, 861, 1393, 2108, 3033, 4195, 5621, ...;
  1, 239, 1632, 5827, 15200, 32856, 62629, 109082, 177507, 273925,...;
  1, 5828, 68457, 342382, 1144664, 3013980, 6769672, 13570796, ...;
  1, 342383, 7112055, 50110483, 215155493, 690729981, 1828979530, ...;
  1, 50110484, 1879090014, 18757984045, 103674882878, 406279238154,..;
  1, 18757984046, 1287814075131, 18318289003447, 130648799730635, ...;
Triangle A078122 begins:
  1;
  1,     1;
  1,     3,      1;
  1,    12,      9,     1;
  1,    93,    117,    27,    1;
  1,  1632,   3033,  1080,   81,   1;
  1, 68457, 177507, 86373, 9801, 243, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078122 equals column 3^k - 1 of this table A125800.
Matrix square A078122^2 begins:
     1;
     2,     1;
     5,     6,     1;
    23,    51,    18,    1;
   239,   861,   477,   54,   1;
  5828, 32856, 25263, 4347, 162, 1; ...
where row sums form column 2 of this table A125790,
and column 0 of A078122^2 forms column 1 of this table A125790.

Robert Israel, Table of n, a(n) for n = 0..2484 (antidiagonals 0 to 69, flattened)

Cf. A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal), A125805 (antidiagonal sums); related table: A125800 (q=2).

f[0]:= 1/(1-z):
S[0]:= series(f[0],z,21):
for n from 1 to 20 do
  ff:= unapply(f[n-1],z);
  f[n]:= simplify(1/3*sum(ff(w*z^(1/3)),w=RootOf(Z^3-1,Z)))/(1-z);
  S[n]:= series(f[n],z,21-n)
od:
seq(seq(coeff(S[s-i],z,i),i=0..s),s=0..20); # Robert Israel, Jun 02 2019

T[0, ] = T[, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)

T(n,k,p=0,q=3)=local(A=Mat(1), B); if(n

T(n,k) = T(n,k-1) + T(n-1,3*k) for n > 0, k > 0, with T(0,n)=T(n,0)=1 for n >= 0.

G.f. of row n is g_n(z) where g_{n+1}(z) = (1-z)^(-1)*Sum_{w^3=1} g_n(w*z^(1/3)) (the sum being over the cube roots of unity). - Robert Israel, Jun 02 2019

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

Original entry on oeis.org

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]))

A078123 Square of infinite lower triangular matrix A078122.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 23, 51, 18, 1, 239, 861, 477, 54, 1, 5828, 32856, 25263, 4347, 162, 1, 342383, 3013980, 3016107, 699813, 39285, 486, 1, 50110484, 690729981, 865184724, 253656252, 19053063, 354051, 1458, 1, 18757984046, 406279238154

Offset: 0

Paul D. Hanna, Nov 18 2002

			Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:
[1,_0,_0,0]^2=[_1,_0,_0,_0]
[1,_1,_0,0]___[_2,_1,_0,_0]
[1,_3,_1,0]___[_5,_6,_1,_0]
[1,12,_9,1]___[23,51,18,_1]

Alois P. Heinz, Rows n = 0..60, flattened

Cf. A078121, A078122, A078124, A078125.

S:= proc(i, j) option remember;
       add(M(i, k)*M(k, j), k=0..i)
    end:
M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(S(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(S(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015

S[i_, j_] := S[i, j] = Sum[M[i, k]*M[k, j], {k, 0, i}]; M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[S[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[S[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

M(1, j) = A078125(j), M(j+1, j)=2*3^j.

A078121 Infinite lower triangular matrix, M, that satisfies [M^2](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

Original entry on oeis.org

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, 1, 1828, 8148, 10816, 5456, 1024, 64, 1, 1, 27338, 167568, 274856, 174336, 43680, 4096, 128, 1, 1, 692004, 5866452, 11622976, 8909648, 2794496, 349504, 16384, 256, 1

Offset: 0

Paul D. Hanna, Nov 18 2002

M also satisfies: [M^(2k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(2^n)](i,j) = M(i+n,j+n) for all n>=0.

			The square of the matrix is the same matrix excluding the first row and column:
  [1, 0, 0, 0, 0]^2 = [ 1, 0, 0, 0, 0]
  [1, 1, 0, 0, 0]     [ 2, 1, 0, 0, 0]
  [1, 2, 1, 0, 0]     [ 4, 4, 1, 0, 0]
  [1, 4, 4, 1, 0]     [10,16, 8, 1, 0]
  [1,10,16, 8, 1]     [36,84,64,16, 1]

Alois P. Heinz, Rows n = 0..80, flattened
MathOverflow, Closed form for diagonals of A078121, (2025).

Cf. A002577, A016131, A078122, A089177.

M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(M(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(M(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015

rows = 10; M[k_] := Table[ Which[j == 1, 1, i == j, 1, 1 < j < i, m[i, j], True, 0], {i, 1, k}, {j, 1, k}]; m2[i_, j_] := m[i+1, j+1]; M2[k_] := Table[ Which[jJean-François Alcover, Feb 27 2015 *)
M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[M[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[M[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 27 2015, after Alois P. Heinz *)

rows_upto(n) = my(A, v1); v1 = vector(n+1, i, vector(i, j, 0)); v1[1][1] = 1; for(i=1, n, v1[i+1][1] = 1; v1[i+1][i+1] = 1); for(i=2, n, for(j=1, i-1, A = (i+j+1)%2; v1[i+1][j+1] = 2*sum(k=0, (i-j-1)\2, v1[i-j+1][2*k+A+1]*v1[j+2*k+A+1][j]))); v1 \\ Mikhail Kurkov, Aug 27 2025

M(1,j) = A002577(j) (partitions of 2^j into powers of 2), M(j+1,j) = 2^j, M(j+2,j) = 4^j, M(j+3,j) = A016131(j).
M(n,k) = the coefficient of x^(2^n - 2^(n-k)) in the power series expansion of 1/Product_{j=0..n-k} (1-x^(2^j)) whenever 0<=k0 (conjecture).
M(n,k) = Sum_{j=0..n-k-1} M(n-k,j)*M(k+j,k-1)*(1+(-1)^(n+k+j+1)) for 0 < k < n with M(n,0) = M(n,n) = 1. - Mikhail Kurkov, Jun 01 2025
From Mikhail Kurkov, Jul 01 2025: (Start)
Conjecture 1: let R(n,x) be the n-th row polynomial, then R(n,x) = x*R(n-1,x) + Sum_{k=1..n-1} M(n-1,k-1)*R(k,x)*(-1)^(n+k+1) = R(n-1,x) + x*Sum_{k=1..n-1} (M(n-1,k) - M(n-2,k))*R(k,x) for n > 1 with R(0,x) = 1, R(1,x) = x + 1.
Conjecture 2: M(n+m,n) ~ 2^(m*(2*n+m-1)/2)/m! as n -> oo. More generally, it also looks like that M(n+m,n) for m > 0 can be represented as (Sum_{j=0..flooor((m-1)/2)} 2^((m-2*j)*(2*(n-j)+m-1)/2)*P(m,j)*(-1)^j)/m! where P(m,j) are some positive integer coefficients. (End)

A078125 Number of partitions of 3^n into powers of 3.

Original entry on oeis.org

1, 2, 5, 23, 239, 5828, 342383, 50110484, 18757984046, 18318289003448, 47398244089264547, 329030840161393127681, 6190927493941741957366100, 318447442589056401640929570896, 45106654667152833836835578059359839

Offset: 0

Paul D. Hanna, Nov 18 2002

nonn

a(n) = sum of the n-th row of lower triangular matrix of A078122.
From Valentin Bakoev, Feb 22 2009: (Start)
a(n) = the partitions of 3^n into powers of 3.
A125801(n) = a(n+1) - 1.
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)

			Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:
[1,_0,_0,0]^2=[_1,_0,_0,_0]
[1,_1,_0,0]___[_2,_1,_0,_0]
[1,_3,_1,0]___[_5,_6,_1,_0]
[1,12,_9,1]___[23,51,18,_1]
To obtain t_3(5,2) 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,...,162(= k.m^{n-1}). It is: 1,2,3,4,5,6,7,8,...,162; 1,5,12,22,35,51,...,4510; (this row contains the first 55 members of A000326 - the pentagonal numbers) 1,23,93,238,485,...,29773; 1,239,1632,5827,15200,32856,62629; 1,5828,68457; Column 1 contains the first 5 members of this sequence. - _Valentin Bakoev_, Feb 22 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. A078121, A078122 (matrix shift when cubed), A078123, A078124, A125801.
Column k=3 of A145515. - Alois P. Heinz, Sep 27 2011
Cf. A000244, A002577, A145513.

import Data.MemoCombinators (memo2, list, integral)
a078125 n = a078125_list !! n
a078125_list = f [1] where
   f xs = (p' xs $ last xs) : f (1 : map (* 3) xs)
   p' = memo2 (list integral) integral p
   p  0 = 1; p []  = 0
   p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m
-- Reinhard Zumkeller, Nov 27 2015

m[i_, j_] := m[i, j]=If[j==0||i==j, 1, m3[i-1, j-1]]; m2[i_, j_] := m2[i, j]=Sum[m[i, k]m[k, j], {k, j, i}]; m3[i_, j_] := m3[i, j]=Sum[m[i, k]m2[k, j], {k, j, i}]; a[n_] := m2[n, 0]

Denote the sum m^n + m^n + ... + m^n, k times, by k*m^n (m > 1, n > 0 and k are natural numbers). The general formula for the number of all partitions of the sum k*m^n into powers of m is t_m(n, k)= k+1 if n=1, t_m(n, k)= 1 if k=0, and t_m(n, k)= t_m(n, k-1) + t_m(n-1, k*m) if n > 1 and k > 0. a(n) is obtained for m=3 and n=1,2,3,... - Valentin Bakoev, Feb 22 2009

a(n) = [x^(3^n)] 1/Product_{j>=0} (1-x^(3^j)). - Alois P. Heinz, Sep 27 2011

cp's OEIS Frontend

A078124 Second column, M(n+1,1) for n>=0, of infinite lower triangular matrix M defined in A078122.

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A111815 Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A111816 Column 0 of the matrix logarithm (A111815) of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying the element in row n by n!.

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A125800 Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.

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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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A078123 Square of infinite lower triangular matrix A078122.

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