A056920 -id:A056920 - cp's OEIS frontend

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

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

Paul D. Hanna, Mar 16 2005

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).

			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;
  ...

G. C. Greubel, Rows n=0..100 of triangle, flattened

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

/* 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

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 *)

{T(n,k)=(-1)^(n-k)*(n-k)!*binomial(1+n\2,k+1-(n+1)\2)* binomial( (n+1)\2,k-n\2)};

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

A145118 Denominator polynomials for continued fraction generating function for n!.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Oct 02 2008

Row sums are A056920. T(n,1) gives quarter squares A002620. T(n,2) appears to coincide with 2*A000241(n+1).

			Triangle begins:
1;
1;
1,  -1;
1,  -2;
1,  -4,   2;
1,  -6,   6;
1,  -9,  18,    -6;
1, -12,  36,   -24;
1, -16,  72,   -96,   24;
1, -20, 120,  -240,  120;
1, -25, 200,  -600,  600,  -120;
1, -30, 300, -1200, 1800,  -720;
1, -36, 450, -2400, 5400, -4320, 720;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A021010, A008297, A066667, A105278.

T:= (n, k)-> (-1)^k* binomial(iquo(n+1, 2),k) *binomial(iquo(n, 2), k)*k!:
seq (seq (T(n, k), k=0..iquo(n, 2)), n=0..16);  # Alois P. Heinz, Dec 04 2012

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

cp's OEIS Frontend

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

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