A271875 Triangle T(n,m) = Sum_{k=1..n-m} (k*(-1)^k*binomial(m+k-1,k)*binomial(2*(n-m),n-m-k))/(n-m), with 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..125, flattened

Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.

Formula

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

T(n,m) = Sum_{k=1..n-m}((k*(-1)^k*binomial(m+k-1,k)*binomial(n,n-m-k)*binomial(2*n-2*m+k-1,n-m))/(n-m+k)), T(n,n)=1.

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

T(n,m) = -1 if n=2*m otherwise -m/(n-2*m)*binomial(2*n-3*m-1,n-m). - Vladimir Kruchinin, Jul 22 2025

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

T[n_, m_]:= T[n, m]=Sum[(k*(-1)^k*Binomial[m+k-1, k]*Binomial[2*(n-m), n-m-k])/(n-m), {k, 1, n-m}]; Flatten[Table[If[n==k, 1, T[n, k]], {n, 1, 11}, {k, 1, n}]] (* Indranil Ghosh, Feb 20 2017 *)

Prog

(Maxima)
T(n, m):=if n=m then 1 else sum(k*(-1)^k*binomial(m+k-1, k)*binomial(2*(n-m), n-m-k), k, 1, n-m)/(n-m);
T(n, m):=if n=m then 1 else sum((k*(-1)^k*binomial(m+k-1, k)*binomial(n, n-m-k)*binomial(2*n-2*m+k-1, n-m))/(n-m+k), k, 1, n-m);
taylor(-(2*x^2*y)/(2*x^2*y+sqrt(1-4*x)-1), x, 0, 7, y, 0, 7);
(PARI) T(n, m) = if(n==2*m, -1, -m/(n-2*m)*binomial(2*n-3*m-1, n-m)); \\ Seiichi Manyama, Jul 23 2025

Crossrefs

Cf. A000108, A271825.

Sequence in context: A138224 A181472 A271825 * A324194 A046205 A046206

Adjacent sequences: A271872 A271873 A271874 * A271876 A271877 A271878

Keyword

sign,tabl

Author

Vladimir Kruchinin, Apr 16 2016

Status

approved