A271825 Triangle read by rows: T(n,m) = (-1)^(n-m-1)*m*binomial(2*n-3*m-1,n-m-1)/(n-m), T(n,n)=1.

1, 1, 1, -1, 2, 1, 2, -1, 3, 1, -5, 2, 0, 4, 1, 14, -5, 1, 2, 5, 1, -42, 14, -3, 0, 5, 6, 1, 132, -42, 9, -1, 0, 9, 7, 1, -429, 132, -28, 4, 0, 2, 14, 8, 1, 1430, -429, 90, -14, 1, 0, 7, 20, 9, 1, -4862, 1430, -297, 48, -5, 0, 0, 16, 27, 10, 1

Offset

1,5

Links

Indranil Ghosh, Rows 1..100, flattened

Formula

G.f.: -(x*sqrt(4*x+1)*y+x*y)/(x*sqrt(4*x+1)*y+x*y-2).

Sum_{n>=m) T(n,m)*x^n is expansion of (x*(1+sqrt(1+4*x))/2)^m.

Example

Triangle begins
    1;
    1,  1;
   -1,  2,  1;
    2, -1,  3,  1;
   -5,  2,  0,  4,  1;
   14, -5,  1,  2,  5,  1;
  -42, 14, -3,  0,  5,  6,  1;

Mathematica

Flatten[Table[If[n==m, 1, (-1)^(n-m-1)*m*Binomial[2n-3m-1, n-m-1]/(n-m)], {n, 1, 11}, {m, 1, n}]] (* Indranil Ghosh, Feb 28 2017 *)

Prog

(Maxima)
taylor(1/(1-y*A(x))-1, x, 0, 10, y, 0, 10);
(PARI) a(n, m)=if(n==m, 1, (-1)^(n-m-1)*m*binomial(2*n-3*m-1, n-m-1)/(n-m));
tabl(nn)=for(n=1, nn, for(m=1, n, print1(a(n, m), ", ")); print)
tabl(11); \\ Indranil Ghosh, Feb 28 2017

Crossrefs

Cf. A000108.

Sequence in context: A138222 A138224 A181472 * A271875 A324194 A046205

Adjacent sequences: A271822 A271823 A271824 * A271826 A271827 A271828

Keyword

sign,tabl

Author

Vladimir Kruchinin, Apr 14 2016

Status

approved