A144505 Triangle read by rows: coefficients of polynomials arising from the recurrence A[n](x) = A[n-1]'(x)/(1-x) with A[0] = exp(x).

1, 1, -1, 2, 1, -5, 7, -1, 9, -30, 37, 1, -14, 81, -229, 266, -1, 20, -175, 835, -2165, 2431, 1, -27, 330, -2330, 9990, -24576, 27007, -1, 35, -567, 5495, -34300, 137466, -326515, 353522, 1, -44, 910, -11522, 97405, -561386, 2148139, -4976315, 5329837

Offset: 0

Author

N. J. A. Sloane, Dec 14 2008

Keywords

• sign
• tabf

Examples

			The first few polynomials P[n] (n >= 0) are:
  P[0] = 1;
  P[1] = 1;
  P[2] = -x +2;
  P[3] =  x^2 -5*x +7;
  P[4] = -x^3 + 9*x^2 - 30*x +37;
  P[5] =  x^4 -14*x^3 + 81*x^2 - 229*x +266;
  P[6] = -x^5 +20*x^4 -175*x^3 + 835*x^2 -2165*x +2431;
  P[7] =  x^6 -27*x^5 +330*x^4 -2330*x^3 +9990*x^2 -24576*x +27007;
...
Triangle of coefficients begins:
   1;
   1;
  -1,   2;
   1,  -5,    7;
  -1,   9,  -30,     37;
   1, -14,   81,   -229,    266;
  -1,  20, -175,    835,  -2165,    2431;
   1, -27,  330,  -2330,   9990,  -24576,   27007;
  -1,  35, -567,   5495, -34300,  137466, -326515,   353522;
   1, -44,  910, -11522,  97405, -561386, 2148139, -4976315, 5329837;
...
		

Links

• Seiichi Manyama, Rows n = 0..140, flattened
• Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See pp. 42, 63.
• N. J. A. Sloane, Rows 0 through 25 of the triangle, together with the corresponding polynomials P[n](x).

Crossrefs

Columns give A001515 (really A144301), A144498, A001514, A144506, A144507.

Row sums give A001147.

Alternating row sums give A043301.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50);
f:= func< n,x | x^n*(&+[Binomial(n,j)*Factorial(n+j)*(1-1/x)^(n-j)/(2^j*Factorial(n)) : j in [0..n]]) >;
T:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[1] cat [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 02 2023

Maple

A[0]:=exp(x);
P[0]:=1;
for n from 1 to 12 do
A[n]:=sort(simplify( diff(A[n-1],x)/(1-x)));
P[n]:=sort(simplify(A[n]*(1-x)^(2*n-1)/exp(x)));
t1:=simplify(x^(degree(P[n],x))*subs(x=1/x,P[n]));
t2:=series(t1,x,2*n+3);
lprint(P[n]);
lprint(seriestolist(t2));
od:

Mathematica

f[n_, x_]:= x^n*Sum[((n+j)!/((n-j)!*j!*2^j))*(1-1/x)^(n-j), {j,0,n}];
t[n_, k_]:= Coefficient[Series[f[n,x], {x,0,30}], x, k];
Join[{1}, Table[t[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Oct 02 2023 *)

SageMath

P. = PowerSeriesRing(QQ, 50)
def f(n,x): return x^n*sum(binomial(n,j)*rising_factorial(n+1,j)*(1-1/x)^(n-j)/2^j for j in range(n+1))
def T(n,k): return P( f(n,x) ).list()[k]
[1] + flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 02 2023

Formula

Let A[0](x) = exp(x), A[n](x) = A[n-1]'(x)/(1-x) for n>0 and let P[n](x) = A[n](x)*(1-x)^(2n-1)/exp(x). Row n of triangle gives coefficients of P[n](x) with exponents of x in decreasing order.

From Vladeta Jovovic, Dec 15 2008: (Start)

P[n] = Sum_{k=0..n} ((n+k)!/((n-k)!*k!*2^k)) * (1-x)^(n-k).

E.g.f.: exp((1-x)*(1-sqrt(1-2*y)))/sqrt(1-2*y). (End)