A078121 -id:A078121 - cp's OEIS frontend

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

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

A078122 Infinite lower triangular matrix, M, that satisfies [M^3](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, 3, 1, 1, 12, 9, 1, 1, 93, 117, 27, 1, 1, 1632, 3033, 1080, 81, 1, 1, 68457, 177507, 86373, 9801, 243, 1, 1, 7112055, 24975171, 15562314, 2371761, 88452, 729, 1, 1, 1879090014, 8786827629, 6734916423, 1291958181, 64392813, 796797, 2187, 1

Offset: 0

Paul D. Hanna, Nov 18 2002

M also satisfies: [M^(3k)](i,j) = [M^k](i+1,j+1) for all i,j,k >=0; thus [M^(3^n)](i,j) = M(i+n,j+n) for all n >= 0.

Conjecture: the sum of the n-th row equals the number of partitions of 3^n into powers of 3 (A078125).

			The cube of the matrix is the same matrix excluding the first row and column:
  [1, 0, 0, 0]^3 = [ 1,  0, 0, 0]
  [1, 1, 0, 0]     [ 3,  1, 0, 0]
  [1, 3, 1, 0]     [12,  9, 1, 0]
  [1,12, 9, 1]     [93,117,27, 1]

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

Cf. A078121, A078123, A078124, A078125, A016142.

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(M(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 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}]; Flatten[Table[m[i, j], {i, 0, 8}, {j, 0, i}]]

M(1, j) = A078124(j), M(j+1, j)=3^j, M(j+2, j) = A016142(j).

M(n, k) = the coefficient of x^(3^n - 3^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(3^j)) whenever 0<=k0 (conjecture).

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

Original entry on oeis.org

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).

A125791 a(n) = 2^(n(n-1)(n-2)/6) for n>=1.

Original entry on oeis.org

1, 1, 2, 16, 1024, 1048576, 34359738368, 72057594037927936, 19342813113834066795298816, 1329227995784915872903807060280344576, 46768052394588893382517914646921056628989841375232

Offset: 1

Paul D. Hanna, Dec 10 2006

nonn

a(n) is a tetrahedral power of 2; exponents of 2 in a(n) begin: 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ..., n*(n-1)*(n-2)/6, ... (cf. A000292).
Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.
Also number of distinct instances of the one-in-three monotone 3SAT problem for n variables. - Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008
Hankel transform of aerated 2-Catalan numbers (A015083). [Paul Barry, Dec 15 2010]

Michael De Vlieger, Table of n, a(n) for n = 1..28
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.

Cf. A125790, A078121; A002577, A000292.

seq(2^(binomial(n, n-3)), n=1..10); # Zerinvary Lajos, Jun 16 2007 [modified by Georg Fischer, Nov 09 2023]

A125791[n_]:=2^Binomial[n,n-3];Array[A125791,15] (* Paolo Xausa, Nov 05 2023 *)

PARI
```
a(n)=if(n<1,0,2^(n*(n-1)*(n-2)/6))
```

/* As determinant of n X n matrix: */
{a(n)=local(q=2,A=Mat(1), B); for(m=1, n, 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(matdet(matrix(n,n,r,c,(A^c)[r,1])))}
for(n=1,15,print1(a(n),", "))

% This generates all 3SAT problem instances
test:-test(4).
test(Max):-
between(1,Max,N),
nl,
one_in_three_monotone_3sat(N,Pss),
write(N:Pss),nl,
fail
; nl.
% generates all one-in-three monotone 3SAT problems involving N variables
one_in_three_monotone_3sat(N,Pss):-
ints(1,N,Is),
findall(Xs,ksubset(3,Is,Xs),Xss),
subset_of(Xss,Pss).
% subset generator
subset_of([],[]).
subset_of([X|Xs],Zs):-
subset_of(Xs,Ys),
add_element(X,Ys,Zs).
add_element(_,Ys,Ys).
add_element(X,Ys,[X|Ys]).
% subsets of K elements
ksubset(0,_,[]).
ksubset(K,[X|Xs],[X|Rs]):-K>0,K1 is K-1,ksubset(K1,Xs,Rs).
ksubset(K,[_|Xs],Rs):-K>0,ksubset(K,Xs,Rs).
% list of integers in [From..To]
ints(From,To,Is):-findall(I,between(From,To,I),Is).
% Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008

Determinant of n X n upper left corner submatrix of table A125790.

a(n) = 2^(binomial(n,n-3)). - Zerinvary Lajos, Jun 16 2007, modified to reflect the new offset by Paolo Xausa, Nov 06 2023.

Name simplified; determinant formula moved out of name into formula section by Paul D. Hanna, Oct 16 2013

Offset changed to 1 by Paolo Xausa, Nov 06 2023

A078536 Infinite lower triangular matrix, M, that satisfies [M^4](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, 4, 1, 1, 28, 16, 1, 1, 524, 496, 64, 1, 1, 29804, 41136, 8128, 256, 1, 1, 5423660, 10272816, 2755264, 130816, 1024, 1, 1, 3276048300, 8220685104, 2804672704, 178301696, 2096128, 4096, 1, 1, 6744720496300, 21934062166320, 9139625620672, 729250931456, 11442760704, 33550336, 16384, 1

Offset: 0

Paul D. Hanna, Nov 29 2002

M also satisfies: [M^(4k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(4^n)](i,j) = M(i+n,j+n) for all n>=0. Conjecture: sum of the n-th row equals the partitions of 4^n into powers of 4.

			The 4th power of matrix is the same matrix excluding the first row and column:
[1,__0,__0,_0,0]^4=[____1,____0,___0,__0,0]
[1,__1,__0,_0,0]___[____4,____1,___0,__0,0]
[1,__4,__1,_0,0]___[___28,___16,___1,__0,0]
[1,_28,_16,_1,0]___[__524,__496,__64,__1,0]
[1,524,496,64,1]___[29804,41136,8128,256,1]

Cf. A078121, A078122, A078537.

dim = 9;
a[, 0] = 1; a[i, i_] = 1; a[i_, j_] /; j > i = 0;
M = Table[a[i, j], {i, 0, dim-1}, {j, 0, dim-1}];
M4 = MatrixPower[M, 4];
sol = Table[M4[[i, j]] == M[[i+1, j+1]], {i, 1, dim-1}, {j, 1, dim-1}] // Flatten // Solve;
Table[a[i, j], {i, 0, dim-1}, {j, 0, i}] /. sol // Flatten (* Jean-François Alcover, Oct 20 2019 *)

M(n, k) = the coefficient of x^(4^n - 4^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(4^j)) whenever 0<=k0 (conjecture).

More terms from Jean-François Alcover, Oct 20 2019

A098539 Lower triangular matrix T, read by rows, that shifts left one column under the matrix square of T, with T(n,0)=T(n,1) for n>0 and T(n,n)=1 for n>=0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 26, 26, 20, 8, 1, 166, 166, 140, 72, 16, 1, 1626, 1626, 1460, 888, 272, 32, 1, 25510, 25510, 23884, 16392, 6256, 1056, 64, 1, 664666, 664666, 639156, 479736, 215696, 46816, 4160, 128, 1, 29559718, 29559718, 28895052

Offset: 0

Paul D. Hanna, Sep 13 2004

Column 0 forms A002449, the number of different types of binary trees of height n. Row sums form A098540. Column 1 equals A098541. As a matrix, T satisfies [T^2](n,k) = T(n+1,k+1) for all n,k>=0, where [T^2] denotes the matrix square of T, with T(0,k)=[T^2](k,0) and T(k,k)=1 for all k>=0. This is a variant of triangle A078121.

			Rows of T begin:
[1],
[1,1],
[2,2,1],
[6,6,4,1],
[26,26,20,8,1],
[166,166,140,72,16,1],
[1626,1626,1460,888,272,32,1],
[25510,25510,23884,16392,6256,1056,64,1],
[664666,664666,639156,479736,215696,46816,4160,128,1],...
Matrix square T^2 begins:
[1],
[2,1],
[6,4,1],
[26,20,8,1],
[166,140,72,16,1],
[1626,1460,888,272,32,1],...
showing that T^2 is the same as T after shifting left one column.

Cf. A002449, A098540, A078121.

T(n,k)=local(A,B,C,m);A=matrix(1,1);A[1,1]=1; for(m=2,n+1,B=A^2;C=matrix(m,m);for(i=1,m, for(j=1,i, if(i<3 || j==i || j>m-1,C[i,j]=1,if(j==1,C[i,j]=B[i-1,1],C[i,j]=B[i-1,j-1]));)); A=C);A[n+1,k+1]

T(n, 0) = A002449(n), T(n, n)=1 for n>=0; T(n, 1)=T(n, 0) for n>0.

A111825 Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 96, 36, 1, 1, 6306, 3816, 216, 1, 1, 1883076, 1625436, 139536, 1296, 1, 1, 2700393702, 3121837776, 360839016, 5036256, 7776, 1, 1, 19324893252552, 28794284803908, 4200503990976, 78293629296, 181382976, 46656, 1

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = the partitions of (6^n - 6^(n-k)) into powers of 6 <= 6^(n-k).

			Let q=6; the g.f. of column k of matrix power P^m is:
1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
where L(x) satisfies:
x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
and L(x) = x - 4/2!*x^2 + 42/3!*x^3 + 7296/4!*x^4 +... (A111829).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(6*x) + m^3/3!*L(x)*L(6*x)*L(6^2*x) +
m^4/4!*L(x)*L(6*x)*L(6^2*x)*L(6^3*x) + ...
Triangle P begins:
1;
1,1;
1,6,1;
1,96,36,1;
1,6306,3816,216,1;
1,1883076,1625436,139536,1296,1;
1,2700393702,3121837776,360839016,5036256,7776,1; ...
where P^6 shifts columns left and up one place:
1;
6,1;
96,36,1;
6306,3816,216,1; ...

Cf. A111826 (column 1), A111827 (row sums), A111828 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111830 (q=7), A111835 (q=8).

PARI
```
P(n,k,q=6)=local(A=Mat(1),B);if(n
				
```

Let q=6; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111829).

A125797 Main diagonal of table A125790.

Original entry on oeis.org

1, 2, 9, 84, 1625, 64350, 5174449, 841185704, 275723872209, 181906966455026, 241258554545388985, 642662865556736504700, 3436011253857466940820073, 36852501476559726217536067974, 792571351187806816558255494473185

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; A125798 (diagonal).

a(n)=local(q=2,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]))

A125798 A diagonal of table A125790: a(n) = A125790(n+1,n).

Original entry on oeis.org

1, 4, 35, 656, 25509, 2026564, 326603719, 106355219008, 69808185542089, 92203545302072964, 244779396712068825067, 1305009502037405316440848, 13963029918525356899170492525, 299675759834305402824238609624548

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; A125797 (diagonal).

a(n)=local(q=2,A=Mat(1), B); for(m=1, n+2, 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+1,(A^n)[n+2,c+1]))

A111820 Triangle P, read by rows, that satisfies [P^5](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(5*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 55, 25, 1, 1, 2055, 1525, 125, 1, 1, 291430, 311525, 38875, 625, 1, 1, 165397680, 239305275, 40338875, 975625, 3125, 1, 1, 390075741430, 735920617775, 157056792000, 5077475625, 24409375, 15625, 1

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = the partitions of (5^n - 5^(n-k)) into powers of 5 <= 5^(n-k).

			Let q=5; the g.f. of column k of matrix power P^m is:
1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
where L(x) satisfies:
x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
and L(x) = x - 3/2!*x^2 + 16/3!*x^3 + 2814/4!*x^4 +... (A111824).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(5*x) + m^3/3!*L(x)*L(5*x)*L(5^2*x) +
m^4/4!*L(x)*L(5*x)*L(5^2*x)*L(5^3*x) + ...
Triangle P begins:
1;
1,1;
1,5,1;
1,55,25,1;
1,2055,1525,125,1;
1,291430,311525,38875,625,1;
1,165397680,239305275,40338875,975625,3125,1; ...
where P^5 shifts columns left and up one place:
1;
5,1;
55,25,1;
2055,1525,125,1;
291430,311525,38875,625,1; ...

Cf. A111821 (column 1), A111822 (row sums), A111823 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111825 (q=6), A111830 (q=7), A111835 (q=8).

PARI
```
P(n,k,q=5)=local(A=Mat(1),B);if(n
				
```

Let q=5; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111824).

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

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A078122 Infinite lower triangular matrix, M, that satisfies [M^3](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.

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

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A125791 a(n) = 2^(n*(n-1)*(n-2)/6) for n>=1.

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A078536 Infinite lower triangular matrix, M, that satisfies [M^4](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.

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A098539 Lower triangular matrix T, read by rows, that shifts left one column under the matrix square of T, with T(n,0)=T(n,1) for n>0 and T(n,n)=1 for n>=0.

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A111825 Triangle P, read by rows, that satisfies [P^6](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(6*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0.

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A125791 a(n) = 2^(n(n-1)(n-2)/6) for n>=1.