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.

Showing 1-10 of 21 results. Next

A091352 Row sums of triangle A091351, in which the k-th column lists the row sums of the k-th power of A091351 (when considered as a lower triangular matrix).

Original entry on oeis.org

1, 2, 4, 9, 24, 77, 295, 1329, 6934, 41351, 278680, 2101434, 17574552, 161740316, 1626733108, 17771416521, 209739328924, 2661301094008, 36148700652163, 523597247829867, 8059284921781892, 131408547139817541
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2004

Keywords

Comments

Equals column 1 of table A125781. Equals row sums and column 0 (shifted) of triangle A127420. - Paul D. Hanna, Feb 11 2007

Links

Crossrefs

Cf. A091351, A091353, A091354.
Cf. A125781, A127420.

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

Views

Author

Paul D. Hanna, Jan 02 2004

Keywords

Comments

Also equals the second column of triangle A091351.

Crossrefs

Cf. A091351, A091352, A091354.

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

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

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

Crossrefs

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

Formula

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

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.

A136220 Triangle P, read by rows, 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 row up, with P(0,0)=1.

Original entry on oeis.org

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, 3247546, 2647628, 742620, 122350, 14330, 1325, 105, 8, 1, 65762269, 55251994, 15624420, 2571620
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2007, corrected Jan 24 2008

Keywords

Examples

			Triangle P 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;
   3247546,  2647628,   742620,  122350,  14330,  1325,  105,   8, 1;
  65762269, 55251994, 15624420, 2571620, 298240, 26943, 2030, 136, 9, 1; ...
where column k of P = column 0 of U^(k+1) and U = A136228.
Matrix cube, W = P^3 (A136231), begins:
       1;
       3,     1;
      15,     6,     1;
     108,    48,     9,    1;
    1036,   495,    99,   12,   1;
   12569,  6338,  1323,  168,  15,  1;
  185704, 97681, 21036, 2754, 255, 18, 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 row up.
Matrix square, P^2 (A136225), begins:
      1;
      2,     1;
      8,     4,    1;
     49,    26,    6,    1;
    414,   232,   54,    8,   1;
   4529,  2657,  629,   92,  10,  1;
  61369, 37405, 9003, 1320, 140, 12, 1; ...
where column k of P^2 = column 0 of V^(k+1) and
triangle V = A136230 begins:
      1;
      2,     1;
      8,     5,     1;
     49,    35,     8,    1;
    414,   325,    80,   11,   1;
   4529,  3820,   988,  143,  14,  1;
  61369, 54800, 14696, 2200, 224, 17, 1; ...
where column k of V = column 0 of P^(3k+2).
Related triangle U = A136228 begins:
      1;
      1,     1;
      3,     4,    1;
     15,    24,    7,    1;
    108,   198,   63,   10,   1;
   1036,  2116,  714,  120,  13,  1;
  12569, 28052, 9884, 1725, 195, 16, 1; ...
where column k of U = column 0 of P^(3k+1)
and column k of P = column 0 of U^(k+1).
Surprisingly, column 0 of P is also found in square A136217:
(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.

Crossrefs

Columns: A136221, A136222, A136223, A136224.
Related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.
Variants: A091351, A135880.

Programs

  • PARI
    {T(n,k)=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,k+1]}

Formula

Denote this triangle by P and define as follows.
Let [P^m]_k denote column k of matrix power P^m,
so that triangular matrix W = A136231 may be defined by
[W]_k = [P^(3k+3)]_0, for k>=0, such that
(1) W = P^3 and (2) [W]_0 = [P]_0 shift up one row.
Define the triangular matrix U = A136228 by
[U]_k = [P^(3k+1)]_0, for k>=0,
and the triangular matrix V = A136230 by
[V]_k = [P^(3k+2)]_0, for k>=0.
Then columns of P may be formed from powers of U:
[P]_k = [U^(k+1)]_0, for k>=0,
and columns of P^2 may be formed from powers of V:
[P^2]_k = [V^(k+1)]_0, for k>=0.
Further, columns of powers of P, U, V and W satisfy:
[U^(j+1)]_k = [P^(3k+1)]_j,
[V^(j+1)]_k = [P^(3k+2)]_j,
[W^(j+1)]_k = [P^(3k+3)]_j,
[W^(j+1)]_k = [W^(k+1)]_j,
[P^(3j+3)]_k = [P^(3k+3)]_j, for all j>=0, k>=0.
Also, we have the column transformations:
U * [P]k = [P]{k+1},
V * [P^2]k = [P^2]{k+1},
W * [P^3]k = [P^3]{k+1},
W * [U]k = [U]{k+1},
W * [V]k = [V]{k+1},
W * [W]k = [W]{k+1}, for all k>=0.
Other identities include the matrix products:
U = P * [P^2 shift right one column];
V = P^2 * [P shift right one column];
V = U * [U shift down one row];
W = V * [V shift down one row];
where the triangle transformations "shift right" and "shift down" are illustrated in examples of entries A136228 (U) and A136230 (V).

Extensions

Typo in example corrected by Paul D. Hanna, Mar 27 2010

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

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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}

Formula

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.

A113355 Triangle T, read by rows, equal to the matrix square of triangle A113350, where T transforms column k of T into column k+1 of T.

Original entry on oeis.org

1, 4, 1, 18, 8, 1, 112, 68, 12, 1, 965, 712, 150, 16, 1, 10957, 9270, 2184, 264, 20, 1, 156699, 147174, 37523, 4912, 410, 24, 1, 2727793, 2786270, 754171, 104476, 9280, 588, 28, 1, 56306695, 61662544, 17502145, 2531004, 235025, 15672, 798, 32, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

Also, T transforms column k of A113340^2 into column k+1 of A113340^2. Column 0: T(n,0) = A113356(n) = A113346(n+1) - 1, where A113346 equals column 0 of triangle A113345 (=A113340^2).

Examples

			Triangle T begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1;
156699,147174,37523,4912,410,24,1;
2727793,2786270,754171,104476,9280,588,28,1;
56306695,61662544,17502145,2531004,235025,15672,798,32,1; ...
where T transforms column k of T into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] = [1,12,150,2184,37523,...].

Crossrefs

Cf. A113340, A113350, A113356 (column 0), A113357 (column 1), A113358 (column 2), A113359 (column 3); A091351.

Programs

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

Formula

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) = A113346(n+1) - 1, for n>=0.

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

Views

Author

Paul D. Hanna, Mar 07 2005

Keywords

Comments

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.

Examples

			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; ...

Crossrefs

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

Programs

  • PARI
    T(n,k)=if(n

Formula

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).

A113394 Triangle, read by rows, equal to the matrix cube of triangle A113389.

Original entry on oeis.org

1, 9, 1, 99, 18, 1, 1569, 360, 27, 1, 34344, 9051, 783, 36, 1, 980487, 284148, 26820, 1368, 45, 1, 34930455, 10865358, 1089126, 59250, 2115, 54, 1, 1502349459, 494019714, 51784137, 2946456, 110715, 3024, 63, 1, 76058669082, 26168502684
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Examples

			Triangle A113389^3 begins:
1;
9,1;
99,18,1;
1569,360,27,1;
34344,9051,783,36,1;
980487,284148,26820,1368,45,1;
34930455,10865358,1089126,59250,2115,54,1;
1502349459,494019714,51784137,2946456,110715,3024,63,1;
76058669082,26168502684,2840586075,167137110,6510780,185589,4095,72,1;

Crossrefs

Cf. A113389, A113395 (column 0); recurrence: A091351, A113355.

Programs

  • PARI
    T(n,k)=local(A,B);A=Mat(1);for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(3*j-2))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(3*c))[r-c+1,1]))^3)[n+1,k+1]

Formula

Column k of A113389^3 = column 0 of A113389^(3*k+3) for k>=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

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Crossrefs

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

Formula

a(n) = A091352(n+1) - 1.
Showing 1-10 of 21 results. Next

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