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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A104558 Triangle, read by rows, equal to the matrix inverse of A104557 and related to Laguerre polynomials.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 6, -6, 1, 0, 0, -6, 18, -9, 1, 0, 0, 0, -24, 36, -12, 1, 0, 0, 0, 24, -96, 72, -16, 1, 0, 0, 0, 0, 120, -240, 120, -20, 1, 0, 0, 0, 0, -120, 600, -600, 200, -25, 1, 0, 0, 0, 0, 0, -720, 1800, -1200, 300, -30, 1, 0, 0, 0, 0, 0, 720, -4320, 5400, -2400, 450, -36, 1, 0, 0, 0, 0, 0, 0, 5040, -15120, 12600, -4200, 630, -42, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 16 2005

Keywords

Comments

Even-indexed rows are found in A066667 (generalized Laguerre polynomials). Odd-indexed rows are found in A021009 (Laguerre polynomials L_n(x)). Row sums equal A056920 (offset 1). Absolute row sums equal A056953 (offset 1).

Examples

			Rows begin:
  1;
  -1,1;
  0,-2,1;
  0,2,-4,1;
  0,0,6,-6,1;
  0,0,-6,18,-9,1;
  0,0,0,-24,36,-12,1;
  0,0,0,24,-96,72,-16,1;
  0,0,0,0,120,-240,120,-20,1;
  0,0,0,0,-120,600,-600,200,-25,1;
  ...
Unsigned columns read downwards equals rows of matrix inverse A104557 read backwards:
  1;
  1,1;
  2,2,1;
  6,6,4,1;
  24,24,18,6,1;
  120,120,96,36,9,1;
  ...

Links

Crossrefs

Cf. A104557, A066667, A021009, A056920, A056953.

Programs

  • Magma
    /* As triangle */ [[(-1)^(n-k)*Factorial(n-k)*Binomial(1+ Floor(n/2), k +1 -Floor((n+1)/2))*Binomial(Floor((n+1)/2), k - Floor(n/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 14 2018
  • Mathematica
    T[n_, k_] := (-1)^(n - k)*(n - k)!*Binomial[1 + Floor[n/2], k + 1 - Floor[(n + 1)/2]]*Binomial[Floor[(n+1)/2], k -Floor[n/2]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2018 *)
  • PARI
    {T(n,k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2,k+1-(n+1)\2)* binomial( (n+1)\2,k-n\2)};

Formula

T(n, k) = (-1)^(n-k)*(n-k)!*C(1+[n/2], k+1-[(n+1)/2])*C([(n+1)/2], k-[n/2]).

A104559 Triangle, read by rows, of the number of left factors of peakless Motzkin paths of length n having k number of U's and D's (i.e., number of paths from (0,0) to the line x=n, consisting of steps U=(1,1), H=(1,0), D=(1,1), that never go below the x-axis and a U step is never followed by a D step).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 6, 1, 1, 5, 16, 18, 9, 1, 1, 6, 25, 40, 36, 12, 1, 1, 7, 36, 75, 100, 60, 16, 1, 1, 8, 49, 126, 225, 200, 100, 20, 1, 1, 9, 64, 196, 441, 525, 400, 150, 25, 1, 1, 10, 81, 288, 784, 1176, 1225, 700, 225, 30, 1, 1, 11, 100, 405, 1296, 2352
Offset: 0

Views

Author

Paul D. Hanna and Emeric Deutsch, Mar 16 2005

Keywords

Comments

Row sums form A091964, the number of left factors of peakless Motzkin paths of length n.

Examples

			Triangle begins:
  1;
  1,   1;
  1,   2,   1;
  1,   3,   4,   1;
  1,   4,   9,   6,   1;
  1,   5,  16,  18,   9,   1;
  1,   6,  25,  40,  36,  12,   1;
  1,   7,  36,  75, 100,  60,  16,   1;
  1,   8,  49, 126, 225, 200, 100,  20,   1; ...

Crossrefs

Cf. A091964, A104557.

Programs

  • Maple
    T:=proc(n,k) if k<=n then binomial(n-floor(k/2),floor((k+1)/2))*binomial(n-floor((k+1)/2),floor(k/2)) else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Mar 16 2005
  • PARI
    T(n,k)=binomial(n-(k\2),(k+1)\2)*binomial(n-((k+1)\2),k\2)
  • PARI
    {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( 2/(1-X+X^2*Y^2-2*X*Y+sqrt((1-X+X^2*Y^2)^2-4*X^2*Y^2)),n,x),k,y)}

Formula

G.f.: A(x, y) = 2/(1-x+x^2*y^2 - 2*x*y + sqrt((1-x+x^2*y^2)^2 - 4*x^2*y^2)) (due to Emeric Deutsch).
T(n, k) = C(n-floor(k/2), floor((k+1)/2))*C(n-floor((k+1)/2), floor(k/2)) = A104557(n, k)/(n-k)!.

A145316 L-matrix for Hankel matrix of aerated factorials.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 1, 0, 6, 0, 6, 0, 1, 6, 0, 18, 0, 9, 0, 1, 0, 24, 0, 36, 0, 12, 0, 1, 24, 0, 96, 0, 72, 0, 16, 0, 1, 0, 120, 0, 240, 0, 120, 0, 20, 0, 1, 120, 0, 600, 0, 600, 0, 200, 0
Offset: 0

Views

Author

Paul Barry, Oct 07 2008

Keywords

Comments

Row sums are A056953. Hankel matrix of aerated factorials H=LDU where U=transpose of L. Aerated version of A104557.

Examples

			Triangle begins 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 1, 0, 6, 0, 6, 0, 1, 6, 0, 18, 0, 9, 0, 1, 0, 24, 0, 36, 0, 12, 0, 1, 24, 0, 96, 0, 72, 0, 16, 0, 1

Formula

Triangle T(n,k)=[k<=n]((n-k)/2)!C((n+k)/2-floor(k/2),floor((k+1)/2))C((n+k)/2-floor((k+1)/2),floor(k/2))(1+(-1)^(n-k))/2;
Showing 1-3 of 3 results.

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

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