A158706 Expansion of e.g.f.: exp(t*x)/(1 - x/t - t^2 * x^2).

1, 1, 0, 1, 2, 0, 2, 0, 3, 6, 0, 6, 0, 15, 0, 7, 24, 0, 24, 0, 84, 0, 52, 0, 37, 120, 0, 120, 0, 540, 0, 380, 0, 485, 0, 141, 720, 0, 720, 0, 3960, 0, 3000, 0, 5430, 0, 2406, 0, 1111, 5040, 0, 5040, 0, 32760, 0, 26040, 0, 60690, 0, 32802, 0, 28147, 0, 5923

Offset: 0

Author

Roger L. Bagula, Mar 24 2009

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
     1;
     1, 0,    1;
     2, 0,    2, 0,     3;
     6, 0,    6, 0,    15, 0,     7;
    24, 0,   24, 0,    84, 0,    52, 0,    37;
   120, 0,  120, 0,   540, 0,   380, 0,   485, 0,   141;
   720, 0,  720, 0,  3960, 0,  3000, 0,  5430, 0,  2406, 0,  1111;
  5040, 0, 5040, 0, 32760, 0, 26040, 0, 60690, 0, 32802, 0, 28147, 0, 5923;
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A011973, A110313 (row sums).

Programs

Mathematica

(* First program *)
Table[CoefficientList[n!*t^n*SeriesCoefficient[Series[Exp[t*x]/(1 -x/t -t^2*x^2), {x,0,20}], n], t], {n,0,10}]//Flatten
(* Second program *)
Table[CoefficientList[Series[Sum[Sum[GegenbauerC[k, (s+1)/2 -k, 1]*x^(4*k+2*n - 2*s)*(n!/(n-s)!), {k,0,Floor[s/2]}], {s,0,n}], {x,0,20}], x], {n,0,10}] (* G. C. Greubel, Nov 30 2021 *)

Sage

@CachedFunction
def A011973(n,k): return 0 if (k<0 or k>(n//2)) else binomial(n-k, k)
def f(n,x): return sum( sum( (A011973(s,j)/factorial(n-s))*x^(4*j+2*n-2*s) for j in (0..(s//2)) ) for s in (0..n) )
def A158706(n,k): return factorial(n)*( f(n,x) ).series(x,2*n+1).list()[k]
flatten([[A158706(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Nov 30 2021

Formula

T(n, k) = coefficients of the expansion : p(x,t) = exp(t*x)/(1 - x/t - t^2* x^2).

T(n, k) = coefficients of the series : Sum_{s=0..n} Sum_{j=0..floor(s/2)} (n!/(n-s)!)*A011973(s, j)*x^(4*j+2*n-2*s). - G. C. Greubel, Nov 30 2021

Extensions

Edited by G. C. Greubel, Nov 30 2021