A166553 Triangle read by rows: T(n, k) = [x^k]( (n+2)!*(3*EulerE(n, x+1) - EulerE(n, x))/4 ).

1, 3, 3, 0, 24, 12, -30, 0, 180, 60, 0, -720, 0, 1440, 360, 2520, 0, -12600, 0, 12600, 2520, 0, 120960, 0, -201600, 0, 120960, 20160, -771120, 0, 3810240, 0, -3175200, 0, 1270080, 181440, 0, -61689600, 0, 101606400, 0, -50803200, 0, 14515200, 1814400

Offset: 0

Author

Roger L. Bagula, Dec 12 2010

Keywords

• sign
• tabl

Comments

I think the rows are indexed by t = 0, 1, 2, ..., and in each row we expand the polynomial in powers of x. - N. J. A. Sloane, Dec 14 2010

Former name: Triangle read by rows: expansion of p(x,t) = exp(x*t)*(3*exp(t) - 1)/(exp(t) + 1), with coefficient of x^n scaled by multiplication by (n!*(n + 2)!/4). - G. C. Greubel, Nov 30 2024

Examples

			Triangle begins as:
        1;
        3,      3;
        0,     24,      12;
      -30,      0,     180,      60;
        0,   -720,       0,    1440,      360;
     2520,      0,  -12600,       0,    12600,   2520;
        0, 120960,       0, -201600,        0, 120960,   20160;
  -771120,      0, 3810240,       0, -3175200,      0, 1270080, 181440;
		

Links

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

Crossrefs

Cf. A001715, A005990, A028296.

Programs

Magma

m:= 13;
R:=PowerSeriesRing(Integers(), m+1);
EulerE:= func< n | (2^(n+1)/(n+1))*( Evaluate(BernoulliPolynomial(n+1), 1/2) - 2^(n+1)*Evaluate(BernoulliPolynomial(n+1), 1/4) ) >;
f:= func< n,x | (Factorial(n+2)/2)*( 3*x^n - 2*(&+[ Binomial(n,j)*(EulerE(j)/2^j)*(x - 1/2)^(n-j): j in [0..n]]) ) >;
A166553:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A166553(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Nov 30 2024

Mathematica

(* first program *)
p[t_]= Exp[x*t](3*Exp[t] - 1)/(Exp[t] + 1);
With[{m=12}, Table[(n!*(n+2)!/2)*CoefficientList[SeriesCoefficient[ Series[p[t], {t,0,m+1}], n], x], {n,0,m}]]//Flatten
(* Second program *)
f[n_, x_]:= (n+2)!*(3*EulerE[n, x+1] - EulerE[n, x])/4;
A166553[n_, k_]:= Coefficient[Series[f[n, x], {x,0,n}], x, k];
Table[A166553[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 30 2024 *)

SageMath

def f(n,x): return (factorial(n+2)/2)*( 3*x^n - 2*sum( binomial(n,j)*euler_number(j)*(x-1/2)^(n-j)/2^j for j in range(n+1)) )
def A166553(n,k): return ( f(n,x) ).series(x,n+1).list()[k]
print(flatten([[A166553(n,k) for k in range(n+1)] for n in range(14)])) # G. C. Greubel, Nov 30 2024

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (n!*(n+2)!/2) * [t^n]( exp(x*t)*(3*exp(t) - 1)/(exp(t) + 1) ).

From G. C. Greubel, Nov 30 2024: (Start)

T(n, k) = [x^k]( (n+2)!*(3*EulerE(n, x+1) - EulerE(n, x))/4 ).

T(n, k) = [x^k]( (1/2)*(n+2)!*( 3*x^n - 2*Sum_{j=0..n} binomial(n,j)*(EulerE(j)/2^j)*(x - 1/2)^(n-j) ) ).

T(n, n) = 3*A001715(n+2) = (n+2)!/2.

T(n, n-1) = 3*A005990(n+1). (End)

Extensions

I rewrote the definition. - N. J. A. Sloane, Dec 14 2010

New name by G. C. Greubel, Nov 30 2024