cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A091354 Row sums of the matrix cube 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, 4, 16, 70, 344, 1908, 11904, 83028, 642960, 5490560, 51373420, 523581128, 5781166688, 68819889018, 879350377816, 12012238559559, 174794145558664, 2700485871440464, 44163954923956850, 762460145368056070
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2004

Keywords

Comments

Also equals the third column of triangle A091351.

Crossrefs

Cf. A091351, A091352, A091353.

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

Views

Author

Paul D. Hanna, Mar 08 2005

Keywords

Comments

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

Examples

			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; ...

Crossrefs

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

Programs

  • PARI
    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])

Formula

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

Views

Author

Paul D. Hanna, Jan 14 2007

Keywords

Comments

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).

Examples

			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.

Crossrefs

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

Views

Author

Paul D. Hanna, Sep 07 2004

Keywords

Comments

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.

Examples

			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],...

Crossrefs

Cf. A091351, A091352.

Programs

  • PARI
    T(n,k)=if(n

Formula

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

A104447 Column 1 of triangular matrix A104446.

Original entry on oeis.org

1, 2, 5, 13, 39, 139, 587, 2897, 16462, 106301, 771313, 6228073, 55494336, 541651873, 5753940704, 66147591142, 818802488476, 10864622564915, 153914784829775, 2319599022540318, 37068215129072522, 626279667948552452
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2005

Keywords

Comments

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

Crossrefs

Cf. A104446, A091352, A091353.

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+1,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+2,2])

Formula

a(n) = A091352(n-1) + A091353(n-1).

A104448 Row sums of triangle A104446.

Original entry on oeis.org

1, 3, 6, 13, 33, 101, 372, 1624, 8263, 48285, 320031, 2380114, 19675986, 179314868, 1788473424, 19398149629, 227510745445, 2871040422932, 38810001746171, 559745948482030, 8582882169611759, 139467832061599433
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2005

Keywords

Comments

A104446 equals the square of triangular matrix A104445, read by rows, where X=[A104445] satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.

Crossrefs

Cf. A104446, A104445, A091352, A091351.

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+1,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(sum(k=1,n+1,(A^2)[n+1,k]))

Formula

a(n) = A091352(n) + A091352(n-1) for n>0.
Previous Showing 11-16 of 16 results.

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