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 1-4 of 4 results.

A132615 Triangle T, read by rows, where row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 3, 1, 1, 80, 25, 5, 1, 1, 1666, 378, 56, 7, 1, 1, 47232, 8460, 1020, 99, 9, 1, 1, 1694704, 252087, 26015, 2134, 154, 11, 1, 1, 73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1, 3744491970, 420142350, 34461260, 2257413, 125760, 6290, 300, 15, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2007

Keywords

Examples

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  6, 3, 1, 1;
  80, 25, 5, 1, 1;
  1666, 378, 56, 7, 1, 1;
  47232, 8460, 1020, 99, 9, 1, 1;
  1694704, 252087, 26015, 2134, 154, 11, 1, 1;
  73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
GENERATE T FROM ODD MATRIX POWERS OF T.
Matrix cube, T^3, begins:
  1;
  3, 1;
  6, 3, 1; <-- row 3 of T
  31, 12, 3, 1;
  357, 100, 18, 3, 1;
  6786, 1455, 205, 24, 3, 1; ...
where row 3 of T = row 2 of T^3 with appended '1'.
Matrix fifth power, T^5, begins:
  1;
  5, 1;
  15, 5, 1;
  80, 25, 5, 1; <-- row 4 of T
  855, 215, 35, 5, 1;
  15171, 3065, 410, 45, 5, 1; ...
where row 4 of T = row 3 of T^5 with appended '1'.
Matrix seventh power, T^7, begins:
  1;
  7, 1;
  28, 7, 1;
  161, 42, 7, 1;
  1666, 378, 56, 7, 1; <-- row 5 of T
  28119, 5348, 679, 70, 7, 1; ...
where row 5 of T = row 4 of T^7 with appended '1'.
ALTERNATE GENERATING METHOD.
Row 4: start with a '1' followed by 4 zeros;
take partial sums and append 2 zeros; then
take partial sums thrice more:
  (1), 0, 0, 0, 0;
  1, 1, 1, 1, (1), 0, 0;
  1, 2, 3, 4, 5, 5, (5);
  1, 3, 6, 10, 15, 20, (25);
  1, 4, 10, 20, 35, 55, (80);
the final nonzero terms form row 4: [80, 25, 5, 1, 1].
Row 5: start with a '1' followed by 6 zeros;
take partial sums and append 4 zeros;
take partial sums and append 2 zeros; then
take partial sums thrice more:
  (1), 0, 0, 0, 0, 0, 0;
  1, 1, 1, 1, 1, 1, (1), 0, 0, 0, 0;
  1, 2, 3, 4, 5, 6, 7, 7, 7, 7, (7), 0, 0;
  1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 56, (56);
  1, 4, 10, 20, 35, 56, 84, 119, 161, 210, 266, 322, (378);
  1, 5, 15, 35, 70, 126, 210, 329, 490, 700, 966, 1288, (1666);
the final nonzero terms form row 5: [1666, 378, 56, 7, 1, 1].
Continuing in this way produces all the rows of this triangle.

Links

Crossrefs

Cf. columns: A132616, A132617, A132618; A132619; variants: A132610, A101479.
Cf. A304190, A304191, A304193.

Programs

  • Maple
    b:= proc(n) option remember;
      Matrix(n, (i,j)-> T(i-1,j-1))^(2*n-3)
    end:
T:= proc(n,k) option remember;
     `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1]))
    end:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 13 2020
  • Mathematica
    b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-3];
T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k + 1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)
  • PARI
    {T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)*(n-2)-(n-j-1)*(n-j-2); A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}

Formula

T(n+1,1) is divisible by 2n-1 for n>=1.

A132616 Column 0 of triangle A132615.

Original entry on oeis.org

1, 1, 1, 6, 80, 1666, 47232, 1694704, 73552752, 3744491970, 218684051648, 14406896813608, 1056681951098592, 85379764462169382, 7534286318509305600, 720884741940337283712, 74330131862002429961712, 8215901579822006354547330, 969069489665924620416715008
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2007

Keywords

Comments

Triangle T = A132615 is generated by odd matrix powers of itself such that row n+1 of T = row n of T^(2n-1) with appended '1' for n >= 0 with T(0,0) = 1.

Examples

			G.f. = 1 + x + x^2 + 6*x^3 + 80*x^4 + 1666*x^5 + 47232*x^6 + ...

Crossrefs

Cf. A132615 (triangle).
Other columns: A132617, A132618, A132619.

Programs

  • Mathematica
    a[ n_, k_: 1] := a[n, k] = If[ n < 2, Boole[n >= 0], Sum[ a[n - 1, i], {i, k + 2 (n - 2)}]]; (* Michael Somos, Nov 29 2016 *)
  • PARI
    {a(n) = my(A = vector(n+1), p); A[1] = 1; for(j=1, n-1, p = (n-1)*(n-2) - (n-j-1)*(n-j-2); A = Vec((Polrev(A) + x * O(x^p)) / (1-x))); A = Vec((Polrev(A) + x * O(x^p)) / (1-x)); A[p+1]}
  • PARI
    {a(n, k=1) = if( n<2, n>=0, sum(i=1, k + 2*n-4, a(n-1, i)))}; /* Michael Somos, Nov 29 2016 */

Formula

From Benedict W. J. Irwin, Nov 29 2016: (Start)
Conjecture: a(n) is described by a series of nested sums,
a(2) = Sum_{i=1..1} 1,
a(3) = Sum_{i=1..1+2} Sum_{j=1..i} 1,
a(4) = Sum_{i=1..1+4} Sum_{j=1..i+2} Sum_{k=1..j} 1,
a(5) = Sum_{i=1..1+6} Sum_{j=1..i+4} Sum_{k=1..j+2} Sum_{l=1..k} 1,
and so on, where 2, 4, 6,... are the even numbers. (End)

A132617 Column 1 of triangle A132615.

Original entry on oeis.org

1, 1, 3, 25, 378, 8460, 252087, 9392890, 420142350, 21927528948, 1307723670020, 87712176219801, 6534001434836758, 535159878432210000, 47792303301224799288, 4621416976280491118910, 481020078381722064175750
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2007

Keywords

Comments

Triangle T=A132615 is generated by odd matrix powers of itself such that row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.

Crossrefs

Cf. A132615 (triangle); other columns: A132616, A132618; A132619.

Programs

  • PARI
    {a(n)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-1, p=n*(n-1)-(n-j)*(n-j-1); A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}

Formula

a(n) is divisible by 2n-1 for n>0; a(n)/(2n-1) = A132619(n).

A132618 Column 2 of triangle A132615.

Original entry on oeis.org

1, 1, 5, 56, 1020, 26015, 855478, 34461260, 1642995124, 90456911140, 5646312067585, 393937815588880, 30374808071994000, 2564601377235725520, 235302361169390146650, 23309583579201438877060
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2007

Keywords

Comments

Triangle T=A132615 is generated by odd matrix powers of itself such that row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.

Crossrefs

Cf. A132615 (triangle); other columns: A132616, A132617; A132619.

Programs

  • PARI
    {a(n)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-1, p=(n+1)*n-(n-j+1)*(n-j); A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}
Showing 1-4 of 4 results.

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