A078536 -id:A078536 - cp's OEIS frontend

A111818 Matrix log of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -2, 4, 0, 2, -8, 16, 0, 840, 8, -32, 64, 0, -76056, 3360, 32, -128, 256, 0, -158761104, -304224, 13440, 128, -512, 1024, 0, 390564896784, -635044416, -1216896, 53760, 512, -2048, 4096, 0, 14713376473366656, 1562259587136, -2540177664, -4867584, 215040, 2048, -8192, 16384, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Column k equals 4^k multiplied by column 0 (A111819) when ignoring zeros above the diagonal.

			Matrix log of A078536, with factorial denominators, begins:
0;
1/1!, 0;
-2/2!, 4/1!, 0;
2/3!, -8/2!, 16/1!, 0;
840/4!, 8/3!, -32/2!, 64/1!, 0;
-76056/5!, 3360/4!, 32/3!, -128/2!, 256/1!, 0;
-158761104/6!, -304224/5!, 13440/4!, 128/3!, -512/2!, 1024/1!, 0;

Cf. A078536, A111819 (column 0), A111845 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).

PARI
```
T(n,k,q=4)=local(A=Mat(1),B);if(n
				
```

T(n, k) = 4^k*T(n-k, 0) = A111819(n-k) for n>=k>=0.

A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, -2, 2, 840, -76056, -158761104, 390564896784, 14713376473366656, -783793232940393380736, -571732910947761663424746240, 603368029500890443054004423520000, 8390120127886533420753746115877557580800

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Let q=4; the g.f. of column k of A078536^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

			A(x) = x - 2/2!*x^2 + 2/3!*x^3 + 840/4!*x^4 - 76056/5!*x^5 +...
where e.g.f. A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(4*x)/2! + A(x)*A(4*x)*A(4^2*x)/3! +
A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...
Let G(x) be the g.f. of A111817 (column 1 of A078536), then
G(x) = 1 + 4*A(x) + 4^2*A(x)*A(4*x)/2! +
4^3*A(x)*A(4*x)*A(4^2*x)/3! +
4^4*A(x)*A(4*x)*A(4^2*x)*A(4^3*x)/4! + ...

Cf. A078536 (triangle), A111817, A111818 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=4)=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(4^j*x)/(j+1).

A111817 Number of partitions of 3*4^n into powers of 4, also equals column 1 of triangle A078536, which shifts columns left and up under matrix 4th power.

Original entry on oeis.org

1, 4, 28, 524, 29804, 5423660, 3276048300, 6744720496300, 48290009081437356, 1221415413140406958252, 110523986015743458745929900, 36150734459755630877180158951596

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Let q=4; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

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

Cf. A078536 (triangle), A002577 (q=2), A078124 (q=3), A111821 (q=5), A111826 (q=6), A111831 (q=7), A111836 (q=8).

a(n,q=4)=local(A=Mat(1),B);if(n<0,0, 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(A[n+2,2]))

a(n) = [x^(3*4^n)] 1/Product_{j>=0}(1-x^(4^j)).

A078537 Number of partitions of 4^n into powers of 4 (without regard to order).

Original entry on oeis.org

1, 2, 6, 46, 1086, 79326, 18583582, 14481808030, 38559135542174, 357934565638890910, 11766678027350761752990, 1387043469046575118555443614, 592264246356176268834689653440926, 923812464024548700407122072128655860126, 5301247577915139769925461060755690116740047262

Offset: 0

Paul D. Hanna, Nov 29 2002

nonn

Conjecture: a(n) = sum of the n-th row of lower triangular matrix A078536.

			a(2) = 6 since partitions of 4^2 into powers of 4 are: [16], [4,4,4,4], [4,4,4,1,1,1,1], [4,4,1,1,1,1,1,1,1,1], [4,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A002577, A078125, A078536.

Column k=4 of A145515.

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/4]]; b = Table[ a[n], {n, 0, 4^9}]; Table[ b[[4^n + 1]], {n, 0, 9}]

a(n) = coefficient of x^(4^n) in power series expansion of 1/[(1-x)(1-x^4)(1-x^16)...(1-x^(4^k))...].

Extended by Robert G. Wilson v, Dec 01 2002

More terms from Alois P. Heinz, Oct 11 2008

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).

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).

A111830 Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*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, 7, 1, 1, 154, 49, 1, 1, 16275, 8281, 343, 1, 1, 9106461, 6558209, 410914, 2401, 1, 1, 28543862991, 27307109501, 2298650515, 20170801, 16807, 1, 1, 521136519414483, 636922972420469, 67522139062441, 790856748801, 988621354

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = partitions of (7^n - 7^(n-k)) into powers of 7 <= 7^(n-k).

			Let q=7; 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 - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 +... (A111834).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(7*x) + m^3/3!*L(x)*L(7*x)*L(7^2*x) +
m^4/4!*L(x)*L(7*x)*L(7^2*x)*L(7^3*x) + ...
Triangle P begins:
1;
1,1;
1,7,1;
1,154,49,1;
1,16275,8281,343,1;
1,9106461,6558209,410914,2401,1;
1,28543862991,27307109501,2298650515,20170801,16807,1; ...
where P^7 shifts columns left and up one place:
1;
7,1;
154,49,1;
16275,8281,343,1; ...

Cf. A111831 (column 1), A111832 (row sums), A111833 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111835 (q=8).

PARI
```
P(n,k,q=7)=local(A=Mat(1),B);if(n
				
```

Let q=7; 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 (A111834).

A111835 Triangle P, read by rows, that satisfies [P^8](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(8*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, 8, 1, 1, 232, 64, 1, 1, 36968, 16192, 512, 1, 1, 35593832, 21928768, 1047040, 4096, 1, 1, 219379963496, 178379459392, 11424946688, 67096576, 32768, 1, 1, 9003699178010216, 9288403489672000, 748093366229504, 5862250172416

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Also P(n,k) = partitions of (8^n - 8^(n-k)) into powers of 8 <= 8^(n-k).

			Let q=8; 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 - 6/2!*x^2 + 142/3!*x^3 + 31800/4!*x^4 +... (A111839).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(8*x) + m^3/3!*L(x)*L(8*x)*L(8^2*x) + m^4/4!*L(x)*L(8*x)*L(8^2*x)*L(8^3*x) + ...
Triangle P begins:
1;
1,1;
1,8,1;
1,232,64,1;
1,36968,16192,512,1;
1,35593832,21928768,1047040,4096,1;
1,219379963496,178379459392,11424946688,67096576,32768,1; ...
where P^8 shifts columns left and up one place:
1;
8,1;
232,64,1;
36968,16192,512,1; ...

Cf. A111836 (column 1), A111837 (row sums), A111838 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111830 (q=7).

PARI
```
P(n,k,q=8)=local(A=Mat(1),B);if(n
				
```

Let q=8; 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 (A111839).

A111845 Triangle P, read by rows, that satisfies [P^4](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(4*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(k,k)=1 and P(k+1,1)=P(k+1,0) for k>=0.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 40, 40, 16, 1, 1040, 1040, 544, 64, 1, 78240, 78240, 48960, 8320, 256, 1, 18504256, 18504256, 13110400, 2878720, 131584, 1024, 1, 14463224448, 14463224448, 11192599808, 2982187520, 180270080, 2099200, 4096, 1

Offset: 0

Paul D. Hanna, Aug 23 2005

Column 0 and column 1 are equal for n>0.

			Let q=4; 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 = 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 + 56/3!*x^3 + 1728/4!*x^4 + ... (A111849).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(4*x) + m^3/3!*L(x)*L(4*x)*L(4^2*x) +
m^4/4!*L(x)*L(4*x)*L(4^2*x)*L(4^3*x) + ...
Triangle P begins:
1;
1,1;
4,4,1;
40,40,16,1;
1040,1040,544,64,1;
78240,78240,48960,8320,256,1;
18504256,18504256,13110400,2878720,131584,1024,1; ...
where P^4 shifts columns left and up one place:
1;
4,1;
40,16,1;
1040,544,64,1;
78240,48960,8320,256,1; ...

Cf. A111846 (column 0), A111847 (row sums), A111848 (matrix log), A111840 (q=3), A078536 (variant).

{P(n,k,q=4) = my(A=Mat(1),B);if(nPaul D. Hanna, Jul 11 2025):
for(n=0,10, for(k=0,n, print1(P(n,k),", ")); print(""))

Let q=4; 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 = Sum_{n>=1} -(-1)^n/n!*Product_{j=0..n-1} L(q^j*x); L(x) equals the g.f. of column 0 of the matrix log of P (A111849).

A111975 Triangle P, read by rows, that satisfies [P^2](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(2*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(k,k)=1 and P(k+2,2)=P(k+2,0) for k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 4, 4, 1, 16, 16, 16, 8, 1, 96, 96, 96, 64, 16, 1, 896, 896, 896, 704, 256, 32, 1, 13568, 13568, 13568, 11776, 5504, 1024, 64, 1, 345088, 345088, 345088, 317952, 178176, 43776, 4096, 128, 1, 15112192, 15112192, 15112192, 14422016

Offset: 0

Paul D. Hanna, Aug 24 2005

Terms of column 0, column 1 and column 2 in row n are equal for n>2.

			Triangle P begins:
1;
1,1;
1,2,1;
4,4,4,1;
16,16,16,8,1;
96,96,96,64,16,1;
896,896,896,704,256,32,1;
13568,13568,13568,11776,5504,1024,64,1;
345088,345088,345088,317952,178176,43776,4096,128,1; ...
where P^2 shifts columns left and up one place:
1;
2,1;
4,4,1;
16,16,8,1;
96,96,64,16,1; ...
The matrix inverse, P^-1, equals signed P:
1;
-1,1;
1,-2,1;
-4,4,-4,1;
16,-16,16,-8,1; ...

Cf. A111976 (column 0), A111977 (row sums), A111978 (matrix log), A098539 (variant), A078536 (variant).

P(n,k,q=2)=local(A=Mat(1),B);if(n2,(A^q)[i-1,2],1), B[i,j]=(A^q)[i-1,j-1]));));A=B);return(A[n+1,k+1]))

The g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*2^k)^n/n! * Product_{j=0..n-1} L(2^j*x) where L(x) is the g.f. of column 0 of the matrix log of P (A111979) and satisfies: x-x^2 = Sum_{j>=1}(1-2^j*x)*Prod_{i=0..j-1}L(2^i*x).

cp's OEIS Frontend

A111818 Matrix log of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A111819 Column 0 of the matrix logarithm (A111818) of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying the element in row n by n!.

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A111817 Number of partitions of 3*4^n into powers of 4, also equals column 1 of triangle A078536, which shifts columns left and up under matrix 4th power.

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A078537 Number of partitions of 4^n into powers of 4 (without regard to order).

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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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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.

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A111830 Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*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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A111835 Triangle P, read by rows, that satisfies [P^8](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(8*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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