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 1-9 of 9 results.

A152400 Triangle T, read by rows, where column k of T = column 0 of matrix power T^(k+1) for k>0, with column 0 of T = unsigned column 0 of T^-1 (shifted).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 86, 45, 15, 4, 1, 645, 318, 99, 24, 5, 1, 5662, 2671, 794, 182, 35, 6, 1, 56632, 25805, 7414, 1636, 300, 48, 7, 1, 633545, 280609, 78507, 16844, 2990, 459, 63, 8, 1, 7820115, 3381993, 926026, 194384, 33685, 5026, 665, 80, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Examples

			Triangle T begins:
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
86, 45, 15, 4, 1;
645, 318, 99, 24, 5, 1;
5662, 2671, 794, 182, 35, 6, 1;
56632, 25805, 7414, 1636, 300, 48, 7, 1;
633545, 280609, 78507, 16844, 2990, 459, 63, 8, 1;
7820115, 3381993, 926026, 194384, 33685, 5026, 665, 80, 9, 1;...
where column k of T = column 0 of T^(k+1) for k>0
and column 0 of T = unsigned column 0 of T^-1 (shifted).
Amazingly, column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
Matrix inverse T^-1 begins:
1;
-1, 1;
-1, -2, 1;
-3, -2, -3, 1;
-14, -7, -3, -4, 1;
-86, -37, -12, -4, -5, 1;
-645, -252, -71, -18, -5, -6, 1;...
where unsigned column 0 of T^-1 = column 0 of T (shifted).
Matrix square T^2 begins:
1;
2, 1;
8, 4, 1;
45, 22, 6, 1;
318, 152, 42, 8, 1;
2671, 1251, 345, 68, 10, 1;
25805, 11869, 3253, 648, 100, 12, 1;
280609, 126987, 34546, 6898, 1085, 138, 14, 1;...
where column 0 of T^2 = column 1 of T,
and column 2 of T^2 = column 1 of T^3.
Matrix cube T^3 begins:
1;
3, 1;
15, 6, 1;
99, 42, 9, 1;
794, 345, 81, 12, 1;
7414, 3253, 798, 132, 15, 1;
78507, 34546, 8679, 1518, 195, 18, 1;
926026, 407171, 103707, 18734, 2565, 270, 21, 1;...
where column 0 of T^3 = column 2 of T,
and column 3 of T^3 = column 2 of T^4.
Matrix power T^4 begins:
1;
4, 1;
24, 8, 1;
182, 68, 12, 1;
1636, 648, 132, 16, 1;
16844, 6898, 1518, 216, 20, 1;
194384, 81218, 18734, 2912, 320, 24, 1;
2476868, 1047638, 249202, 40932, 4950, 444, 28, 1;...
where column 0 of T^4 = column 3 of T,
and column 2 of T^4 = column 3 of T^3.
Related triangle A127714 begins:
1;
1, 1, 1;
1, 2, 2, 3, 3, 3;
1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...
where right border = column 0 of this triangle A152400.

Crossrefs

Cf. columns: A127715, A152401, A152402, A152403, A152404.
Cf. related triangles: A152405, A127714.

Programs

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

Formula

Column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.
Column k: T(n,k) = Sum_{j=0..n-k} T(n-k,j)*T(j+k-1,k-1) for n>=k>0.
Column 0: T(n,0) = Sum_{j=1..n} T(n,j)*T(j-1,0) for n>=0.

A152401 Column 1 of triangle A152400; also, column 1 of square array A152405.

Original entry on oeis.org

1, 2, 8, 45, 318, 2671, 25805, 280609, 3381993, 44658052, 640279918, 9895457035, 163871617543, 2893296199441, 54230087607409, 1075060346670798, 22467666780553040, 493590689966427930, 11369641989418301977
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Comments

Matrix powers of triangle T=A152400 satisfy: column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Links

Crossrefs

Cf. A152400; A127715, A152402, A152403, A152404; A152405.

A127714 Triangle, read by rows of (n+1)*(n+2)/2 terms, generated by the following rule. Start with a single '1' in row n=0; from then on, obtain row n from row n-1 by inserting zeros in row n-1 at positions: {(j+1)*n - j*(j-1)/2 | j=0..n} and then take partial sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 1, 3, 5, 5, 8, 11, 11, 14, 14, 14, 1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86, 1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645, 1, 6, 20, 48, 90, 132, 132, 196, 293, 434, 619, 804, 804
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2007

Keywords

Examples

			Triangle begins:
  1;
  1, 1, 1;
  1, 2, 2, 3, 3, 3;
  1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
  1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;
  1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645;
  1, 6, 20, 48, 90, 132, 132, 196, 293, 434, 619, 804, 804, 1047, 1362, 1749, 2136, 2136, 2609, 3168, 3727, 3727, 4372, 5017, 5017, 5662, 5662, 5662;
  ...
Obtain row n from row n-1 by inserting zeros in row n-1 at positions:
[n,2*n,3*n-1,4*n-3,5*n-6,6*n-10,...,(j+1)*n - j*(j-1)/2,... | j=0..n],
and then take partial sums; illustrated by the following examples.
Obtain row 3 from row 2 by inserting zeros at positions [3,6,8,9],
and then take partial sums:
[1, 2, 2, 0, 3, 3, 0, 3, 0, 0];
[1, 3, 5, 5, 8,11,11,14,14,14];
Obtain row 4 from row 3 by inserting zeros at positions [4,8,11,13,14],
and then take partial sums:
[1, 3, 5, _5, _0, _8, 11, 11, _0, 14, 14, _0, 14, _0, _0];
[1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86].

Crossrefs

Cf. A127715, A127716; A127632, A000108, A009766; variants: A127452, A127420.

Programs

  • PARI
    {T(n,k)=local(t);if(n<0 || k<0 || k>(n+1)*(n+2)/2-1,0, t=(sqrtint((2*n+3)^2-8*(k+1))-1)\2; if(k==0,1,if(issquare((2*n+3)^2-8*(k+1)),T(n,k-1),T(n,k-1)+T(n-1,k-n+t))))} {/* for(n=0,8,for(k=0,(n+1)*(n+2)/2-1,print1(T(n,k),","));print("")) */}

Formula

T(n,n) = A000108(n);
A009766 (Catalan's triangle) forms lower left sub-triangle;
T(n+1,2*n+1) = A127632(n), where g.f. of A127632 is: 2/(1+sqrt(2*sqrt(1-4*x)-1)).
T(n,n*(n+1)/2) = A127716(n).
T(n,(n+1)*(n+2)/2-1) = A127715(n).

A152404 Column 1 of matrix square of triangle A152400; also, column 3 of square array A152405.

Original entry on oeis.org

1, 4, 22, 152, 1251, 11869, 126987, 1508209, 19651299, 278321523, 4253151796, 69700149063, 1218679465845, 22634882689433, 444893598200458, 9223269744306877, 201091581942120957, 4598872183673896769
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Comments

Matrix powers of triangle T=A152400 satisfy:
column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Crossrefs

Cf. A152400; A127715, A152401, A152402, A152403; A152405.

A152405 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {m*(m+1)/2, m>=0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 86, 45, 14, 4, 1, 645, 318, 86, 22, 5, 1, 5662, 2671, 645, 152, 31, 6, 1, 56632, 25805, 5662, 1251, 232, 41, 7, 1, 633545, 280609, 56632, 11869, 2026, 327, 53, 8, 1, 7820115, 3381993, 633545, 126987, 20143, 2991, 457, 66, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Examples

			Table begins:
(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),...;
(3),(8),14,(22),31,41,(53),66,80,95,(112),130,149,169,190,...;
(14),(45),86,(152),232,327,(457),606,775,965,(1202),1464,1752,2067,...;
(86),(318),645,(1251),2026,2991,(4455),6207,8274,10684,(13934),17653,...;
(645),(2671),5662,(11869),20143,30827,(48480),70355,96990,128959,...;
(5662),(25805),56632,(126987),223977,352936,(582183),874664,1240239,...;
(56632),(280609),633545,(1508209),2748448,4438122,(7641111),11831184,...;
(633545),(3381993),7820115,(19651299),36837937,60743909,...; ...
where row n equals the partial sums of row n-1 after removing terms
at positions {m*(m+1)/2, m>=0} (marked by parenthesis in above table).
For example, to generate row 3 from row 2:
[3,8, 14, 22, 31,41, 53, 66,80,95, 112, 130,...]
remove terms at positions {0,1,3,6,10,...}, yielding:
[14, 31,41, 66,80,95, 130,149,169,190, ...]
then take partial sums to obtain row 3:
[14, 45,86, 152,232,327, 457,606,775,965, ...].
Continuing in this way generates all rows of this table.
RELATION TO POWERS OF A SPECIAL TRIANGULAR MATRIX.
Columns 0 and 1 are found in triangle T=A152400, which begins:
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
86, 45, 15, 4, 1;
645, 318, 99, 24, 5, 1;
5662, 2671, 794, 182, 35, 6, 1;
56632, 25805, 7414, 1636, 300, 48, 7, 1; ...
where column k of T = column 0 of matrix power T^(k+1) for k>=0.
Furthermore, matrix powers of triangle T=A152400 satisfy:
column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.
Column 3 of this square array = column 1 of T^2:
1;
2, 1;
8, 4, 1;
45, 22, 6, 1;
318, 152, 42, 8, 1;
2671, 1251, 345, 68, 10, 1;
25805, 11869, 3253, 648, 100, 12, 1; ...
RELATED TRIANGLE A127714 begins:
1;
1, 1, 1;
1, 2, 2, 3, 3, 3;
1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...
where right border = column 0 of this square array.

Crossrefs

Cf. columns: A127715, A152401, A152404.
Cf. related triangles: A152400, A127714.
Cf. variants: A125781, A127054, A136212, A136217, A135876, A135878, A136730.

Programs

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

A152402 Column 2 of triangle A152400.

Original entry on oeis.org

1, 3, 15, 99, 794, 7414, 78507, 926026, 12010188, 169580899, 2586371577, 42336163519, 739814864633, 13739211766050, 270108101356162, 5602446487013365, 122232180232983149, 2797753890784828302
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Comments

Matrix powers of triangle T=A152400 satisfy:
column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Crossrefs

Cf. A152400; A127715, A152401, A152403, A152404; A152405.

A152403 Column 3 of triangle A152400.

Original entry on oeis.org

1, 4, 24, 182, 1636, 16844, 194384, 2476868, 34461956, 519070980, 8405444924, 145502477888, 2679656009072, 52289220500822, 1077289873778412, 23361106832257087, 531744673834247758, 12673569230875132668
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2008

Keywords

Comments

Matrix powers of triangle T=A152400 satisfy:
column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Crossrefs

Cf. A152400; A127715, A152401, A152402, A152404; A152405.

A193560 Augmentation of the Catalan triangle, A009766. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 14, 14, 1, 10, 41, 86, 86, 1, 15, 95, 327, 645, 645, 1, 21, 190, 965, 2991, 5662, 5662, 1, 28, 343, 2410, 10684, 30827, 56632, 56632, 1, 36, 574, 5334, 31969, 128959, 352936, 633545, 633545, 1, 45, 906, 10766, 83860, 449435
Offset: 0

Views

Author

Clark Kimberling, Jul 30 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193560, if the triangle is written as (w(n,k)), then w(n,n)=A127715(n).

Examples

			First 5 rows of A193560:
1
1...1
1...3...3
1...6...14...14
1...10..41...86...86

Crossrefs

Cf. A193091.

Programs

  • Mathematica
    p[n_, k_] := ((n - k + 1)/(n + 1)) (n + k)!/(n!*k!)  (* Catalan triangle, A009766 *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193560 *)
Flatten[Table[v[n], {n, 0, 10}]]

A127716 a(n) = A127714(n, n*(n+1)/2).

Original entry on oeis.org

1, 1, 3, 11, 72, 473, 3727, 39646, 412679, 4765654, 61015451, 937386045, 14126033578, 227901084348, 3941375013541, 72677938745613, 1517730017193292, 31249446089387086, 675658943143264102, 15356377683044270187
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2007

Keywords

Comments

Triangle A127714 has (n+1)*(n+2)/2 terms in row n for n>=0. A127715 forms the rightmost border of triangle A127714.

Crossrefs

Cf. A127714, A127715.
Showing 1-9 of 9 results.

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