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.

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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

A118441 Triangle L, read by rows, equal to the matrix log of A118435, with the property that L^2 consists of a single diagonal (two rows down from the main diagonal).

Original entry on oeis.org

0, 1, 0, -4, 2, 0, -12, 12, 3, 0, 32, -48, -24, 4, 0, 80, -160, -120, 40, 5, 0, -192, 480, 480, -240, -60, 6, 0, -448, 1344, 1680, -1120, -420, 84, 7, 0, 1024, -3584, -5376, 4480, 2240, -672, -112, 8, 0, 2304, -9216, -16128, 16128, 10080, -4032, -1008, 144, 9, 0
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Comments

L = log(A118435) = log(H*[C^-1]*H], where C=Pascal's triangle and H=A118433 where H^2 = I (identity matrix).

Examples

			The matrix log, L = log(H*[C^-1]*H], begins:
     0;
     1,     0;
    -4,     2,      0;
   -12,    12,      3,     0;
    32,   -48,    -24,     4,     0;
    80,  -160,   -120,    40,     5,     0;
  -192,   480,    480,  -240,   -60,     6,     0;
  -448,  1344,   1680, -1120,  -420,    84,     7,   0;
  1024, -3584,  -5376,  4480,  2240,  -672,  -112,   8,  0;
  2304, -9216, -16128, 16128, 10080, -4032, -1008, 144,  9,  0;
  ...
The matrix square, L^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;
  0, 0,  0,  0, 30,  0,  0;
  ...
From _Peter Luschny_, Apr 23 2020: (Start)
In unsigned form and without the main diagonal, as computed by the Maple script:
  [0], [0]
  [1], [1]
  [2], [4,   2]
  [3], [12,  12,   3]
  [4], [32,  48,   24,   4]
  [5], [80,  160,  120,  40,   5]
  [6], [192, 480,  480,  240,  60,  6]
  [7], [448, 1344, 1680, 1120, 420, 84, 7] (End)

Crossrefs

Cf. A118435 (exp(L)), A118442 (column 0), A118443 (row sums), A027471 (unsigned row sums); A118433 (self-inverse triangle), A001815 (column 1?), A001789 (third of column 2?).

Programs

  • Maple
    # Generalized Worpitzky transform of the harmonic numbers.
CL := p -> PolynomialTools:-CoefficientList(expand(p), x):
H := n -> add(1/k, k=1..n):
Trow := proc(n) local k,v; if n=0 then return [0] fi;
add(add((-1)^(n-v)*binomial(k,v)*H(k)*(-x+v-1)^n, v=0..k), k=0..n); CL(%) end:
for n from 0 to 7 do Trow(n) od; # Peter Luschny, Apr 23 2020
  • 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}];
L = MatrixLog[H.Inverse[Cn].H ];
Table[L[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2024 *)
  • PARI
    /* From definition of L as matrix log of H*C^-1*H: */
{L(n,k)=local(H=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))),N=(H*C^-1*H)); Log=sum(p=1,n+1,-(N^0-N)^p/p);Log[n+1,k+1]}
for(n=0, 10, for(k=0, n, print1(L(n, k), ", ")); print(""))
  • PARI
    /* The matrix power L^m is given by: */
{L(n,k,m)=if(m%2==0,if(n==k+m,n!/k!*2^(n-k-m)/(n-k-m)!), if(n>=k+m,n!/k!*2^(n-k-m)/(n-k-m)!*(-1)^(m\2+(n+1)\2-k\2+n-k)))}
for(n=0, 10, for(k=0, n, print1(L(n, k,1), ", ")); print(""))

Formula

For even exponents of L, L^(2m) is a single diagonal:
if n == k+2m, then [L^(2m)](n,k) = n!/k!*2^(n-k-2m)/(n-k-2m)!; else if n != k+2m: [L^(2m)](n,k) = 0.
For odd exponents of L:
if n >= k+2m+1, then [L^(2m+1)](n,k) = n!/k!*2^(n-k-2m-1)/(n-k-2m-1)!*(-1)^(m+[(n+1)/2]-[k/2]+n-k); else if n < k+2m+1: [L^(2m)](n,k) = 0.
Unsigned row sums equals A027471(n+1) = n*3^(n-1).

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
Showing 1-3 of 3 results.

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