OFFSET
0,2
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
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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)
EXAMPLE
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;
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 *)
PROG
(Magma)
m:= 13;
R<x>:=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
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Dec 12 2010
EXTENSIONS
I rewrote the definition. - N. J. A. Sloane, Dec 14 2010
New name by G. C. Greubel, Nov 30 2024
STATUS
approved