A091352 -id:A091352 - cp's OEIS frontend

A091351 Triangle T, read by rows, such that T(n,k) equals the (n-k)-th row sum of T^k, where T^k is the k-th power of T as a lower triangular matrix.

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, 1, 1329, 2602, 1908, 805, 231, 49, 8, 1, 1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1, 1, 41351, 99367, 83028, 39001, 12381, 2919, 540, 81, 10, 1

Offset: 0

Paul D. Hanna, Jan 02 2004

Since T(n,0)=1 for n>=0, then the k-th column of the lower triangular matrix T equals the leftmost column of T^(k+1) for k>=0.

			T(7,3) = 344 = 1*1 + 9*3 + 9*9 + 4*30 + 1*115
= T(4,0)*T(2,2) +T(4,1)*T(3,2) +T(4,2)*T(4,2) +T(4,3)*T(5,2) +T(4,4)*T(6,2).
Rows begin:
{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},
{1,1329,2602,1908,805,231,49,8,1},
{1,6934,15133,11904,5325,1616,364,64,9,1},...

Cf. A091352, A091353, A091354, A104445.

T(n,k)=if(k>n || n<0 || k<0,0,if(k==0 || k==n,1, 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(n, 0)=1 (n>=0).

Equals SHIFT_UP(A104445), or A104445(n+1, k) = T(n, k) for n>=k>=0, where triangular matrix X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.

A125781 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 24, 23, 14, 5, 1, 1, 77, 76, 52, 21, 6, 1, 1, 295, 294, 217, 91, 29, 7, 1, 1, 1329, 1328, 1033, 433, 141, 39, 8, 1, 1, 6934, 6933, 5604, 2307, 739, 216, 50, 9, 1, 1, 41351, 41350, 34416, 13804, 4276, 1274, 306, 62, 10, 1, 1

Offset: 0

Paul D. Hanna, Dec 09 2006

Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,4,8,13,...}:
row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...];
row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];
row 4 = partial sums of [1, 23,52, 141,216,306, 535,695,876,1079,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
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, ...;
1,(4), 8, 14,(21), 29, 39, 50,(62), 75, 90, 106, 123,(141), 160, 181,.;
1,(9), 23, 52,(91), 141, 216, 306,(412), 535, 695, 876, 1079,(1305),..;
1,(24), 76, 217,(433), 739, 1274, 1969,(2845), 3924, 5479, 7335,...;
1,(77), 294, 1033,(2307), 4276, 8200, 13679,(21014), 30534, 45528,...;
1,(295), 1328, 5604,(13804), 27483, 58017, 103545,(167868), 255305,...;
1,(1329), 6933, 34416,(92433), 195978, 451283, 855463,(1454823),...;
1,(6934), 41350, 237328,(688611), 1544074, 3847960, 7700971,...;
1,(41351), 278679, 1822753,(5670713), 13371684, 35818351, 75299744,...;
1,(278680), 2101433, 15473117,(51291468), 126591212, 362337006,...;
1,(2101434), 17574551, 144165763,(506502769), 1303252476,...;
1,(17574552), 161740315, 1464992791,(5430460072), 14517950305,...;
Column 1 of this table equals column 1 of triangle A091351;
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;
1, 1329, 2602, 1908, 805, 231, 49, 8, 1; ...
where column k of A091351 = row sums of matrix power A091351^k for k>=0.

Cf. A091351, A091352; columns: A125782, A125783, A125784, A125785, A125786; diagonals: A125787, A125788; A125789 (antidiagonal sums), A125714.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b+1)/2-2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

Surprisingly, column 1 equals A091352 = column 1 of triangle A091351, in which column k equals row sums of the matrix power A091351^k. Column 3 of this table also equals column 1 of matrix power A091351^2.

A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218.

Original entry on oeis.org

1, 1, 3, 15, 108, 1036, 12569, 185704, 3247546, 65762269, 1515642725, 39211570981, 1125987938801, 35554753133312, 1224882431140838, 45731901253649898, 1839804317195739634, 79355626796692509253, 3653687500034925338348

Offset: 0

Paul D. Hanna, Dec 25 2007

nonn

P = A136220 is a triangular matrix where column k of P^3 equals column 0 of P^(3k+3) such that column 0 of P^3 equals column 0 of P shift one place left. Tables A136217 and A136218 are defined by recurrences seemingly unrelated to the matrix product recurrence of A136220 and yet they all generate this same sequence in column 0.

			Equals column 0 of triangle P=A136220, which begins:
1;
1, 1;
3, 2, 1;
15, 10, 3, 1;
108, 75, 21, 4, 1;
1036, 753, 208, 36, 5, 1;
12569, 9534, 2637, 442, 55, 6, 1;
185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
where column k of P^3 = column 0 of P^(3k+3) such that
column 0 of P^3 = column 0 of P shift one place left.
Surprisingly, column 0 of P is also found in square A136218:
(1),(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),...;
(3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),...;
(15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),...;
(108),(414),1036,(2116),3493,(5555),8040,(11477),15483,...;
(1036),(4529),12569,(28052),48800,(82328),124335,(186261),...;
(12569),(61369),185704,(446560),811111,(1438447),2250731,...;
...
and has a recurrence similar to that of square array A136212
which generates the triple factorials.

Paul D. Hanna, Table of n, a(n) for n = 0..90

Cf. A136220 (P), A136228 (U), A136231 (W=P^3).
Cf. other columns of P: A136222, A136223, A136224.
Cf. related tables: A136217, A136218.
Cf. variants: A091352, A135881.

/* Generate using matrix product recurrences of triangle A136220: */ {a(n)=local(P=Mat(1),U,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));U=P*PShR^2; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));P[n+1,1]}

/* Generated as column 0 in triangle A136218 (faster): */ {a(n)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[0]; for(j=1,#A,if(j+m-1==(m*(m+7))\6,m+=1;B=concat(B,0));B=concat(B,A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));A[1]}

A091353 Row sums of the matrix square of triangle A091351, in which the k-th column lists the row sums of A091351^k (the k-th power of A091351 when considered as a lower triangular matrix).

Original entry on oeis.org

1, 3, 9, 30, 115, 510, 2602, 15133, 99367, 729962, 5949393, 53392902, 524077321, 5592200388, 64520858034, 801031071955, 10654883235991, 151253483735767, 2283450321888155, 36544617881242655, 618220383026770560

Offset: 0

Paul D. Hanna, Jan 02 2004

nonn

Also equals the second column of triangle A091351.

Cf. A091351, A091352, A091354.

A104445 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Mar 07 2005

Surprisingly, SHIFT_UP(T) = A091351, or T(n+1,k) = A091351(n,k) for n>=k>=0, where column k of A091351 equals column 0 of A091351^(k+1) for k>=0.

			Rows begin:
1;
1,1;
1,1,1;
1,2,1,1;
1,4,3,1,1;
1,9,9,4,1,1;
1,24,30,16,5,1,1;
1,77,115,70,25,6,1,1;
1,295,510,344,135,36,7,1,1;
1,1329,2602,1908,805,231,49,8,1,1;
1,6934,15133,11904,5325,1616,364,64,9,1,1; ...

Cf. A091351, A104446 (matrix square); columns form: A091352, A091353, A091354.

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

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

A125782 Column 2 of table A125781.

Original entry on oeis.org

1, 3, 8, 23, 76, 294, 1328, 6933, 41350, 278679, 2101433, 17574551, 161740315, 1626733107, 17771416520, 209739328923, 2661301094007, 36148700652162, 523597247829866, 8059284921781891, 131408547139817540

Offset: 0

Paul D. Hanna, Dec 09 2006

nonn

Cf. A091351; other columns: A091352, A125783, A125784, A125785, A125786.

a(n) = A091352(n+1) - 1.

A125783 Column 3 of table A125781; also, equals column 1 of matrix power A091351^2.

Original entry on oeis.org

1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, 1822753, 15473117, 144165763, 1464992791, 16144683412, 191967912402, 2451561765083, 33487399558154, 487448547177703, 7535687673952024, 123349262218035648

Offset: 0

Paul D. Hanna, Dec 09 2006

nonn

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

Cf. A091351; other columns: A091352, A125782, A125784, A125785, A125786.

a(n) = A091352(n+2) - A091352(n+1) - 1.

A125784 Column 4 of table A125781.

Original entry on oeis.org

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.

cp's OEIS Frontend

A091351 Triangle T, read by rows, such that T(n,k) equals the (n-k)-th row sum of T^k, where T^k is the k-th power of T as a lower triangular matrix.

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A125781 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).

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A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218.

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A091353 Row sums of the matrix square of triangle A091351, in which the k-th column lists the row sums of A091351^k (the k-th power of A091351 when considered as a lower triangular matrix).

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A104445 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1.

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A125782 Column 2 of table A125781.

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A125783 Column 3 of table A125781; also, equals column 1 of matrix power A091351^2.

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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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