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.

A118442 Column 0 of triangle A118441, which is the matrix log of triangle A118435.

Original entry on oeis.org

0, 1, -4, -12, 32, 80, -192, -448, 1024, 2304, -5120, -11264, 24576, 53248, -114688, -245760, 524288, 1114112, -2359296, -4980736, 10485760, 22020096, -46137344, -96468992, 201326592, 419430400, -872415232, -1811939328, 3758096384, 7784628224, -16106127360, -33285996544
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Links

Crossrefs

Cf. A118441 (triangle), A118443 (row sums); A118435, A001787.

Programs

  • Mathematica
    LinearRecurrence[{0,-8,0,-16},{0,1,-4,-12},40] (* Harvey P. Dale, Dec 29 2014 *)
  • PARI
    {a(n)=polcoeff(x*(1 - 4*x - 4*x^2)/(1 + 4*x^2 +x*O(x^n))^2,n)}

Formula

a(n) = (-1)^floor(n/2)*A001787(n).
G.f.: x*(1 - 4*x - 4*x^2)/(1 + 4*x^2)^2.
E.g.f.: x*(cos(2*x) - sin(2*x)). - Stefano Spezia, Jul 01 2023

A118443 Row sums of triangle A118441, which is the matrix log of triangle A118435.

Original entry on oeis.org

1, -2, 3, -36, -155, 474, 1127, -1992, -1719, -4810, -31669, 109332, 286637, -596974, -904785, 449136, -3218287, 16156782, 50232979, -121747380, -233735691, 309853258, 15768823, 1624290984, 6853579225, -19712646746, -44873974053, 79998871428, 90434035261
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Links

Crossrefs

Cf. A118441 (triangle), A118442 (column 0), A118444 (a(n)/(n+1)); A118435.

Programs

  • Mathematica
    nmax = 30;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
L = MatrixLog[H.Inverse[Cn].H];
Total /@ Rest@L (* Jean-François Alcover, Apr 08 2024 *)
  • PARI
    {a(n)=polcoeff((1+x)*(1-3*x+18*x^2-78*x^3+45*x^4-175*x^5)/(1+6*x^2+25*x^4 +x*O(x^n))^2,n)}

Formula

G.f.: (1+x)*(1-3*x+18*x^2-78*x^3+45*x^4-175*x^5)/(1+6*x^2+25*x^4)^2.
E.g.f.: cos(2*x)*((1 - x)*cosh(x) + (1 + 3*x)*sinh(x)) - sin(2*x)*((1 + x)*cosh(x) - (1 - 3*x)*sinh(x)). - Stefano Spezia, Jul 01 2023

A118444 a(n) = A118443(n)/(n+1), where A118443 is the row sums of triangle A118441.

Original entry on oeis.org

1, -1, 1, -9, -31, 79, 161, -249, -191, -481, -2879, 9111, 22049, -42641, -60319, 28071, -189311, 897599, 2643841, -6087369, -11130271, 14084239, 685601, 67678791, 274143169, -758178721, -1661999039, 2857102551, 3118415009, 1811852719, 22839485921, -82298680089, -214997290751
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Comments

A118441 is the matrix log of triangle A118435.
Given the series S = (1, -i)^n, n>0: (1, -1), (0, -2), (-2, -2), ...; the real part of the binomial transform of S = (1, 1, -1, -9, -31, -79, -161, -249, -191, 481, ...). - Gary W. Adamson, Sep 19 2008

Links

Crossrefs

Cf. A118441, A118443, A118435.

Programs

  • Mathematica
    LinearRecurrence[{0, -6, 0, -25}, {1, -1, 1, -9}, 33] (* Jean-François Alcover, Apr 08 2024 *)
  • PARI
    {a(n)=polcoeff((1-x+13*x^2-21*x^3+67*x^4-115*x^5+175*x^6-375*x^7) /(1+6*x^2+25*x^4 +x*O(x^n))^2,n)}

Formula

G.f.: (1 - x + 13*x^2 - 21*x^3 + 67*x^4 - 115*x^5 + 175*x^6 - 375*x^7) / (1 + 6*x^2 + 25*x^4)^2.
For n > 3, a(n) = 4*a(n-1) - 5*a(n-2). - Gary W. Adamson, Aug 08 2006
E.g.f.: exp(x)*cos(2*x) - sin(2*x)*(cosh(x) - sinh(x)). - Stefano Spezia, Jul 01 2023
a(n) = (-1)^floor((n+1)/2)*(1+i)*((2+i)^n-i*(2-i)^n)/2, where i is the imaginary unit. - Gerry Martens, Mar 31 2024

A118435 Triangle T, read by rows, equal to the matrix product T = H*[C^-1]*H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.

Original entry on oeis.org

1, 1, 1, -3, 2, 1, -11, 15, 3, 1, 25, -44, -18, 4, 1, 41, -115, -110, 50, 5, 1, -43, 246, 375, -220, -45, 6, 1, 29, 315, 861, -805, -385, 105, 7, 1, -335, 232, -1204, 2296, 1750, -616, -84, 8, 1, -1199, 3033, 1044, 3780, 5166, -2898, -924, 180, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Comments

The matrix inverse of H*[C^-1]*H is H*C*H = A118438, where H^2 = I (identity). The matrix log, log(T) = A118441, is a matrix square root of a triangular matrix with a single diagonal (two rows down from the main diagonal).

Examples

			Triangle begins:
  1;
  1, 1;
  -3, 2, 1;
  -11, 15, 3, 1;
  25,-44,-18, 4, 1;
  41,-115,-110, 50, 5, 1;
  -43, 246, 375,-220,-45, 6, 1;
  29, 315, 861,-805,-385, 105, 7, 1;
  -335, 232,-1204, 2296, 1750,-616,-84, 8, 1;
  -1199, 3033, 1044, 3780, 5166,-2898,-924, 180, 9, 1;
  ...
The matrix log, log(T) = A118441, starts:
  0;
  1, 0;
  -4, 2, 0;
  -12, 12, 3, 0;
  32,-48,-24, 4, 0;
  80,-160,-120, 40, 5, 0;
  ...
where matrix square, log(T)^2, is a single diagonal:
  0;
  0,0;
  2,0,0;
  0,6,0,0;
  0,0,12,0,0;
  0,0,0,20,0,0;
  ...

Crossrefs

Cf. A118436 (column 0), A118437 (row sums), A118438 (matrix inverse), A118441 (matrix log), A118433 (self-inverse H).

Programs

  • Mathematica
    nmax = 12;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
Tn = H.Inverse[Cn].H;
T[n_, k_] := Tn[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2024 *)
  • PARI
    {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1)*(-1)^(r\2- (c-1)\2+r-c))),C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))));(M*C^-1*M)[n+1,k+1]}

Formula

Since T + T^-1 = C + C^-1, then [T^-1](n,k) = (1+(-1)^(n-k))*C(n,k) - T(n,k) is a formula for the matrix inverse T^-1 = A118438.

A118438 Triangle T, read by rows, equal to the matrix product T = H*C*H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.

Original entry on oeis.org

1, -1, 1, 5, -2, 1, 11, -9, -3, 1, -23, 44, 30, -4, 1, -41, 125, 110, -30, -5, 1, 45, -246, -345, 220, 75, -6, 1, -29, -301, -861, 875, 385, -63, -7, 1, 337, -232, 1260, -2296, -1610, 616, 140, -8, 1, 1199, -3015, -1044, -3612, -5166, 3150, 924, -108, -9, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Comments

The matrix inverse of H*C*H is H*[C^-1]*H = A118435, where H^2 = I (identity). The matrix log, log(T) = -A118441, is a matrix square root of a triangular matrix with a single diagonal (two rows down from the main diagonal).

Examples

			Triangle begins:
1;
-1, 1;
5,-2, 1;
11,-9,-3, 1;
-23, 44, 30,-4, 1;
-41, 125, 110,-30,-5, 1;
45,-246,-345, 220, 75,-6, 1;
-29,-301,-861, 875, 385,-63,-7, 1;
337,-232, 1260,-2296,-1610, 616, 140,-8, 1;
1199,-3015,-1044,-3612,-5166, 3150, 924,-108,-9, 1; ...

Crossrefs

Cf. A118439 (column 0), A118440 (row sums), A118435 (matrix inverse), A118441 (-matrix log); A118433 (self-inverse H).

Programs

  • Mathematica
    nmax = 12;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
Tn = H.Cn.H;
T[n_, k_] := Tn[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2024 *)
  • PARI
    {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1)*(-1)^(r\2- (c-1)\2+r-c))),C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))));(M*C*M)[n+1,k+1]}

Formula

Since T + T^-1 = C + C^-1, then [T^-1](n,k) = (1+(-1)^(n-k))*C(n,k) - T(n,k) is a formula for the matrix inverse T^-1 = A118435.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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