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 1-5 of 5 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.

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]}

A132619 Column 1 of triangle A132615 divided by twice the row index less 1.

Original entry on oeis.org

1, 1, 5, 54, 940, 22917, 722530, 28009490, 1289854644, 68827561580, 4176770296181, 284087018905946, 21406395137288400, 1770085307452770344, 159359206078637624790, 15516776721991034328250
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); columns: A132616, A132617, A132618.

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]/(2*n-1)}

Formula

a(n) = A132615(n+1,1)/(2n-1) = A132617(n)/(2n-1) for n>=1.

A304190 G.f. A(x) satisfies: [x^n] (1+x)^((n-1)^2) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 0, 0, 4, 90, 2448, 79716, 3058740, 135637242, 6835557984, 386119895256, 24170805494868, 1661105052140226, 124342746871407984, 10070793262851698412, 877493877654988612836, 81848857574562663295026, 8137513480199793111630528, 859067817713438540813133744, 95973644392465888508242272804, 11312379843382901418721437545706
Offset: 0

Views

Author

Paul D. Hanna, May 08 2018

Keywords

Examples

			G.f.: A(x) = 1 + 4*x^3 + 90*x^4 + 2448*x^5 + 79716*x^6 + 3058740*x^7 + 135637242*x^8 + 6835557984*x^9 + 386119895256*x^10 + 24170805494868*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^((n-1)^2) / A(x) begins:
n=0: [1, 1, 0, -4, -94, -2538, -82148, -3137720, ...];
n=1: [1, 0, 0, -4, -90, -2448, -79700, -3058020, ...];
n=2: [1, 1, 0, -4, -94, -2538, -82148, -3137720, ...];
n=3: [1, 4, 6, 0, -105, -2832, -90048, -3391872, ...];
n=4: [1, 9, 36, 80, 0, -3276, -105224, -3871476, ...];
n=5: [1, 16, 120, 556, 1666, 0, -123900, -4673220, ...];
n=6: [1, 25, 300, 2296, 12460, 47232, 0, -5561820, ...];
n=7: [1, 36, 630, 7136, 58671, 368784, 1694704, 0,  ...];
n=8: [1, 49, 1176, 18420, 211590, 1895322, 13604628, 73552752, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n-1)^2) / A(x) = 0 for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 1, 6, 80, 1666, 47232, 1694704, 73552752, ...]
yields A132616, column 0 of triangle A132615.
Related triangular matrix T = A132615 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; ...
in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.

Crossrefs

Cf. A132616, A304191, A304192, A304193, A132615.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^((m-2)^2)/Ser(A) )[m] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

A132616(n+1) = [x^n] (1+x)^(n^2) / A(x) for n>=0.
Showing 1-5 of 5 results.

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