A091351 -id:A091351 - cp's OEIS frontend

A125784 Column 4 of table A125781.

1, 5, 21, 91, 433, 2307, 13804, 92433, 688611, 5670713, 51291468, 506502769, 5430460072, 62894124926, 783259655434, 10445143907067, 148592182641759, 2247301621235992, 36021020633412788, 610161098104988668

Offset: 0

Paul D. Hanna, Dec 09 2006

nonn

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

			a(n) = A125783(n) + A125786(n-1) for n>0:
A125783 begins: 1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, ...
and A125786 begins: 1, 7, 39, 216, 1274, 8200, 58017, 451283, ...
term-by-term addition forms this sequence.
This sequence can also be derived from the matrix square A091351^2:
1;
2, [1];
4, [4, 1];
9, [14, 6, 1];
24, [52, 30, 8, 1];
77, [217, 153, 52, 10, 1];
295, [1033, 845, 336, 80, 12, 1];
1329, [5604, 5152, 2294, 625, 114, 14, 1]; ...
The terms enclosed in square barackets sum to equal this sequence.

Cf. A091351; other columns: A091352, A125782, A125783, A125785, A125786.

a(n) = Sum_{k=0..n} [A091351^2](n+1,k+1) where A091351^2 is the matrix square of A091351.

A125785 Column 5 of table A125781.

Original entry on oeis.org

1, 6, 29, 141, 739, 4276, 27483, 195978, 1544074, 13371684, 126591212, 1303252476, 14517950305, 174196495882, 2241822436160, 30826098464147, 451299846525541, 7012090426122158, 115289977296253757, 2000463474160276658

Offset: 0

Paul D. Hanna, Dec 09 2006

nonn

A091352 equals column 1 of both table A125781 and triangle A091351.

			A091352 begins: [1, 2, 4, 9, 24, 77, 295, 1329, 6934, 41351, ...];
a(5) = A091352(8) - 2*A091352(7) = 6934 - 2*1329 = 4276.

Cf. A125781; other columns: A091352, A125782, A125783, A125784, A125786.

a(n) = A091352(n+2) - 2*A091352(n+1). a(n) = A125782(n+1) - A125783(n+1).

A125786 Column 6 of table A125781.

Original entry on oeis.org

1, 7, 39, 216, 1274, 8200, 58017, 451283, 3847960, 35818351, 362337006, 3965467281, 46749441514, 591291743032, 7993582141984, 115104783083605, 1759853074058289, 28485332959460764, 486811835886953020

Offset: 0

Paul D. Hanna, Dec 09 2006

nonn

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

			a(n) = A125784(n+1) - A125783(n+1) for n>0:
A125784 begins: 1, 5, 21, 91, 433, 2307, 13804, 92433, 688611, ...;
A125783 begins: 1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, ...;
term-by-term differences form this sequence.
This sequence can also be derived from the matrix square A091351^2:
1;
2, 1;
4, 4, [1];
9, 14, [6, 1];
24, 52, [30, 8, 1];
77, 217, [153, 52, 10, 1];
295, 1033, [845, 336, 80, 12, 1];
1329, 5604, [5152, 2294, 625, 114, 14, 1]; ...
the terms enclosed in square barackets sum to equal this sequence.

Cf. A125781; other columns: A091352, A125782, A125783, A125784, A125785.

a(n) = Sum_{k=0..n} [A091351^2](n+2,k+2) where A091351^2 is the matrix square of A091351.

A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0 and where T^k is the k-th power of T as a lower triangular matrix.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 5, -11, 9, -4, 1, -17, 38, -33, 16, -5, 1, 71, -162, 145, -74, 25, -6, 1, -357, 824, -753, 396, -140, 36, -7, 1, 2101, -4892, 4535, -2434, 885, -237, 49, -8, 1, -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1, 108609, -255824, 241621, -133012, 50001, -13992, 3073, -548, 81

Offset: 0

Paul D. Hanna, Dec 20 2004

Column 0 forms A101900. Absolute row sums form A101901.

			Rows begin:
        1;
       -1,     1;
        1,    -2,      1;
       -2,     4,     -3,     1;
        5,   -11,      9,    -4,     1;
      -17,    38,    -33,    16,    -5,    1;
       71,  -162,    145,   -74,    25,   -6,   1;
     -357,   824,   -753,   396,  -140,   36,  -7,   1,
     2101, -4892,   4535, -2434,   885, -237,  49,  -8,  1;
   -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1;
      ...

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

Cf. A091351, A101900, A101901, A101898, A101899.

t[n_, k_] := t[n, k] = If[k>n || n<0 || k<0, 0, If[k==n, 1, If[k==0, -Sum[t[n, j], {j, 1, n}], Sum[t[n-k, j]*t[j+k-1, k-1], {j, 0, n-k}]]]]; Table[t[n ,k], {n,0,10}, {k, 0, n}] //Flatten (* Amiram Eldar, Nov 26 2018 *)

{T(n,k)=if(k>n||n<0||k<0,0,if(k==n,1, if(k==0,-sum(j=1,n,T(n,j)), sum(j=0,n-k,T(n-k,j)*T(j+k-1,k-1));));)}

T(n, k) = Sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n >= k > 0 with T(0, 0) = 1 and T(n, 0) = -Sum_{j=1, n} T(n, j) for n > 0.

A135902 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 102, 47, 15, 4, 1, 860, 356, 102, 24, 5, 1, 8548, 3252, 860, 186, 35, 6, 1, 97094, 34448, 8548, 1736, 305, 48, 7, 1, 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1, 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1

Offset: 0

Paul D. Hanna, Dec 18 2007

This is a variant of permutation triangle P = A094587, A094587(n,k) = n!/k!, which may be defined by: triangular matrix P where column k of P = column 0 of P^(k+1), with column 0 of P = column 0 of P^2 shift right.

			Triangle T begins:
1;
1, 1;
3, 2, 1;
15, 8, 3, 1;
102, 47, 15, 4, 1;
860, 356, 102, 24, 5, 1;
8548, 3252, 860, 186, 35, 6, 1;
97094, 34448, 8548, 1736, 305, 48, 7, 1;
1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1;
17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^3 shift right.
Matrix square of T, T^2, begins:
1;
2, 1;
8, 4, 1;
47, 22, 6, 1;
356, 156, 42, 8, 1;
3252, 1343, 351, 68, 10, 1;
34448, 13493, 3415, 656, 100, 12, 1; ...
where column 0 of T^2 = column 1 of T.
Matrix cube of T, T^3, begins:
1;
3, 1;
15, 6, 1;
102, 42, 9, 1;
860, 351, 81, 12, 1;
8548, 3415, 807, 132, 15, 1;
97094, 37795, 8967, 1530, 195, 18, 1; ...
where column 0 of T^3 = column 2 of T = column 0 of T shift left;
also, column 1 of T^3 = column 2 of T^2.

Cf. columns: A135903, A135904, A135905; variants: A091351, A094587.

T(n, k)=if(k>n || n<0 || k<0, 0, if(k==n,1,if(k==0, T(n+1,2), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

A138271 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^4 shift right.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 268, 78, 18, 4, 1, 3164, 798, 156, 28, 5, 1, 43672, 9874, 1714, 268, 40, 6, 1, 682632, 141282, 22368, 3164, 420, 54, 7, 1, 11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1, 224283416, 40400466, 5566728, 682632, 77720

Offset: 0

Paul D. Hanna, Mar 11 2008

			Triangle T begins:
1;
1, 1;
4, 2, 1;
28, 10, 3, 1;
268, 78, 18, 4, 1;
3164, 798, 156, 28, 5, 1;
43672, 9874, 1714, 268, 40, 6, 1;
682632, 141282, 22368, 3164, 420, 54, 7, 1;
11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1;
224283416, 40400466, 5566728, 682632, 77720, 8378, 868, 88, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^4 shift right:
column 1 of T = column 0 of T^2;
column 2 of T = column 0 of T^3;
column 3 of T = column 0 of T^4.
Matrix square of T, T^2, begins:
1;
2, 1;
10, 4, 1;
78, 26, 6, 1;
798, 232, 48, 8, 1;
9874, 2578, 486, 76, 10, 1;
141282, 33764, 5888, 864, 110, 12, 1;
2273730, 503910, 82210, 11396, 1390, 150, 14, 1; ...
where column k of T^2 = column 1 of T^(k+1):
column 0 of T^2 = column 1 of T;
column 2 of T^2 = column 1 of T^3;
column 3 of T^2 = column 1 of T^4.
Matrix cube of T, T^3, begins:
1;
3, 1;
18, 6, 1;
156, 48, 9, 1;
1714, 486, 90, 12, 1;
22368, 5888, 1050, 144, 15, 1;
333910, 82210, 14046, 1908, 210, 18, 1;
5566728, 1289928, 211182, 28072, 3120, 288, 21, 1; ...
where column k of T^3 = column 2 of T^(k+1):
column 0 of T^3 = column 2 of T;
column 1 of T^3 = column 2 of T^2;
column 3 of T^3 = column 2 of T^4.
Matrix 4th power of T, T^4, begins:
1;
4, 1;
28, 8, 1;
268, 76, 12, 1;
3164, 864, 144, 16, 1;
43672, 11396, 1908, 232, 20, 1;
682632, 170000, 28072, 3520, 340, 24, 1;
11834536, 2814832, 454848, 57408, 5820, 468, 28, 1; ...
where column k of T^4 = column 3 of T^(k+1):
column 0 of T^4 = column 3 of T = column 0 of T shift left;
column 1 of T^4 = column 3 of T^2;
column 2 of T^4 = column 3 of T^3.

Cf. columns: A138272, A138273, A138274; central terms: A138275; variants: A091351, A094587, A135902.

{T(n, k) = if(k>n||k<0, 0, if(k==n, 1, if(k==0, T(n+2, 3), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* Column k of T = column 0 of T^(k+1): */
{T(n, k) = local(M=if(n==0,Mat(1),matrix(n,n,r,c,if(r>=c,T(r-1,c-1))))); if(k==n, 1, if(k==0, (M^4)[n, 1],(M^(k+1))[n-k+1, 1]))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

A104446 Square of triangular matrix A104445, read by rows, where X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 10, 13, 7, 2, 1, 25, 39, 25, 9, 2, 1, 78, 139, 100, 41, 11, 2, 1, 296, 587, 459, 205, 61, 13, 2, 1, 1330, 2897, 2418, 1149, 366, 85, 15, 2, 1, 6935, 16462, 14506, 7233, 2421, 595, 113, 17, 2, 1, 41352, 106301, 98161, 50905, 17706, 4535, 904

Offset: 0

Paul D. Hanna, Mar 08 2005

Column 0: T(n,0) = 1 + A091352(n-1) for n>0. Column 1 is A104447. Row sums form A104448.

			Rows begin:
1;
2,1;
3,2,1;
5,5,2,1;
10,13,7,2,1;
25,39,25,9,2,1;
78,139,100,41,11,2,1;
296,587,459,205,61,13,2,1;
1330,2897,2418,1149,366,85,15,2,1
6935,16462,14506,7233,2421,595,113,17,2,1; ...

Cf. A104445, A091351, A091352, A104447, A104448.

T(n,k)=local(A=Mat(1),B);for(m=1,n,B=A^2-A+A^0; A=matrix(m+1,m+1);for(i=1,m+1, for(j=1,i, if(i<2 || j==i,A[i,j]=1,if(j==1,A[i,j]=1,A[i,j]=B[i-1,j-1]))))); return((A^2)[n+1,k+1])

T(n, k) = A104445(n, k) + A104445(n+1, k+1) - I(n, k), where I=identity matrix. T(n, k) = A091351(n-1, k) + A091351(n, k+1) - I(n, k), for n>k>=0.

A127420 Triangle, read by rows, where row n+1 is generated from row n by first inserting zeros at positions {(m+2)*(m+3)/2, m>=0} in row n and then taking the partial sums in reverse order, for n>=2, starting with 1's in the initial two rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 9, 5, 5, 3, 1, 1, 24, 15, 15, 10, 5, 5, 2, 1, 77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1, 295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1, 1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1, 6934, 5605

Offset: 0

Paul D. Hanna, Jan 14 2007

Column 0 forms A091352, which also equals column 1 of table A125781, where table A125781 is generated by a complementary recurrence of this triangle. The number of terms in row n is A127419(n).

			To generate row 6, start with row 5:
24, 15, 15, 10, 5, 5, 2, 1;
insert zeros at positions [1,4,8,13,..., (m+2)*(m+3)/2 - 2,...]:
24, 0, 15, 15, 0, 10, 5, 5, 0, 2, 1;
then row 6 equals the partial sums of row 5 taken in reverse order:
24, _0, 15, 15, _0, 10, _5, 5, 0, 2, 1;
77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1.
Triangle begins:
1;
1, 1;
2, 1, 1;
4, 2, 2, 1;
9, 5, 5, 3, 1, 1;
24, 15, 15, 10, 5, 5, 2, 1;
77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1;
295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1;
1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1;
Column 0 of this triangle equals column 1 of triangle A091351, where triangle A091351 begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 9, 4, 1;
1, 24, 30, 16, 5, 1;
1, 77, 115, 70, 25, 6, 1;
1, 295, 510, 344, 135, 36, 7, 1; ...
and column k of A091351 = row sums of matrix power A091351^k for k>=0.

Cf. A091352, A091351, A125781, A127419.

A098446 Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the (n-1)-th diagonal with the k-th row of this triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 30, 24, 1, 1, 6, 25, 70, 115, 77, 1, 1, 7, 36, 135, 344, 510, 295, 1, 1, 8, 49, 231, 805, 1908, 2602, 1329, 1, 1, 9, 64, 364, 1616, 5325, 11904, 15133, 6934, 1, 1, 10, 81, 540, 2919, 12381, 39001, 83028, 99367, 41351, 1

Offset: 0

Paul D. Hanna, Sep 07 2004

The rows of this triangle are the reverse of the rows of triangle A091351, in which the k-th column lists the row sums of the k-th matrix power of A091351. Row sums form A091352 and equal the secondary diagonal.

			T(7,3) = T(3,0)*T(6,3) + T(3,1)*T(5,2) + T(3,2)*T(4,1) + T(3,3)*T(3,0)
= 1*70 + 3*16 + 4*4 + 1*1 = 135.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,4,1],
[1,4,9,9,1],
[1,5,16,30,24,1],
[1,6,25,70,115,77,1],
[1,7,36,135,344,510,295,1],
[1,8,49,231,805,1908,2602,1329,1],
[1,9,64,364,1616,5325,11904,15133,6934,1],...

Cf. A091351, A091352.

PARI
```
T(n,k)=if(n
				
```

T(n, k) = Sum_{i=0..k} T(k, i)*T(n-i-1, k-i) for 0

A098447 Triangle T, read by rows, such that diagonal n equals column 0 of T^(n+1), the (n+1)-th matrix power of T.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 32, 24, 1, 1, 6, 25, 78, 150, 79, 1, 1, 7, 36, 155, 532, 1018, 340, 1, 1, 8, 49, 271, 1395, 5802, 10996, 2090, 1, 1, 9, 64, 434, 3036, 21343, 116658, 212434, 20613, 1, 1, 10, 81, 652, 5824, 60209, 661325, 5072504

Offset: 0

Paul D. Hanna, Sep 07 2004

Row sums form A098448.

			T(7,3) = T(3,0)*T(3,0) + T(3,1)*T(4,1) + T(3,2)*T(5,2) + T(3,3)*T(6,3)
= 1*1 + 3*4 + 4*16 + 1*78 = 155.
Rows of T begin:
[1],
[1,1],
[1,2,1],
[1,3,4,1],
[1,4,9,9,1],
[1,5,16,32,24,1],
[1,6,25,78,150,79,1],
[1,7,36,155,532,1018,340,1],
[1,8,49,271,1395,5802,10996,2090,1],
[1,9,64,434,3036,21343,116658,212434,20613,1],...
Matrix square T^2 begins:
[1],
[2,1],
[4,4,1],
[9,14,8,1],
[24,53,54,18,1],
[79,234,376,280,48,1],
[340,1291,2976,4034,2196,158,1],...
where column 0 is {1,2,4,9,24,79,340,...} and forms diagonal 1 of T.
Matrix cube T^3 begins:
[1],
[3,1],
[9,6,1],
[32,33,12,1],
[150,219,135,27,1],
[1018,2023,1944,744,72,1],...
where column 0 is {1,3,9,32,150,1018,...} and forms diagonal 2 of T.

Cf. A098448, A098446, A091351.

PARI
```
T(n,k)=if(n
				
```

T(n, k) = Sum_{i=0..k} T(k, i)*T(n-k+i-1, i), for 0

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A125784 Column 4 of table A125781.

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A125785 Column 5 of table A125781.

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A125786 Column 6 of table A125781.

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A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0 and where T^k is the k-th power of T as a lower triangular matrix.

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A135902 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.

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A138271 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^4 shift right.

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A104446 Square of triangular matrix A104445, read by rows, where X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.

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A127420 Triangle, read by rows, where row n+1 is generated from row n by first inserting zeros at positions {(m+2)*(m+3)/2, m>=0} in row n and then taking the partial sums in reverse order, for n>=2, starting with 1's in the initial two rows.

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A098446 Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the (n-1)-th diagonal with the k-th row of this triangle.

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