A121575 Riordan array (-sqrt(4*x^2+8*x+1)+2*x+2, (sqrt(4*x^2+8*x+1)-2*x-1)/2).

1, -2, 1, 6, -5, 1, -24, 24, -8, 1, 114, -123, 51, -11, 1, -600, 672, -312, 87, -14, 1, 3372, -3858, 1914, -618, 132, -17, 1, -19824, 22992, -11904, 4218, -1068, 186, -20, 1, 120426, -140991, 75183, -28383, 8043, -1689, 249, -23, 1, -749976, 884112, -481704, 190347, -58398, 13929, -2508, 321, -26, 1

Offset

0,2

Comments

First column is (-1)^n*A054872(n). Row sums are a signed version of A108524. Inverse of generalized Delannoy triangle A121574. Unsigned triangle is A121576.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, -1, -3, -1, -3, -1, -3, -1, -3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 09 2006

Links

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

Formula

T(n,k) = (-1)^(n-k)*(1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n)*(4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - G. C. Greubel, Nov 02 2018

Example

Triangle begins
     1;
    -2,    1;
     6,   -5,    1;
   -24,   24,   -8,   1;
   114, -123,   51, -11,   1;
  -600,  672, -312,  87, -14, 1;

Mathematica

Flatten[Table[(-1)^(n-k)*Sum[Binomial[n, i] Binomial[2*n-k-i, n]*(4-9*i + 3*i^2 -6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i, {i, 0, n-k}]/2, {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, Nov 02 2018 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1((-1)^(n-k)*sum(j=0, n-k, 2^j*binomial(n, j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ G. C. Greubel, Nov 02 2018
(Magma) [[(-1)^(n-k)*(&+[ 2^j*Binomial(n, j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018
(GAP) T:=Flat(List([0..9], n->List([0..n], k->(-1)^(n-k)*Sum([0..n-k], i->Binomial(n, i)*Binomial(2*n-k-i, n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i)/2))); # Muniru A Asiru, Nov 02 2018

Crossrefs

Sequence in context: A133367 A179456 A214152 * A121576 A049444 A136124

Adjacent sequences: A121572 A121573 A121574 * A121576 A121577 A121578

Keyword

sign,tabl

Author

Paul Barry, Aug 08 2006

Status

approved