A136532 Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.

1, 0, -3, -8, -16, 4, -60, -65, 50, -5, -384, -168, 462, -108, 6, -2380, 763, 3836, -1624, 196, -7, -14208, 21248, 29560, -21472, 4256, -320, 8, -73836, 302571, 199998, -269127, 78840, -9387, 486, -9, -176000, 3761240, 854530, -3288940, 1360150, -228880, 18430, -700, 10

Offset

0,3

Comments

Former title: Coefficients of Laguerre recursive polynomials with an (n+2)!/2 multiplication factor from Hochstadt: P(x, n) = ((2*n - x)*P(x, n-1) - n*P(x, n-2))/(n + 1).

References

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986; pp. 8, 42-43.

Links

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Formula

T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ).

From G. C. Greubel, Jul 25 2023: (Start)

T(n, n) = (-1)^n*(n + 2 - [n=0]).

Sum_{k=0..n} (-1)^k*T(n, k) = A001710(n+2). (End)

Example

Triangle of coefficients begins as:
       1;
       0,     -3;
      -8,    -16,      4;
     -60,    -65,     50,      -5;
    -384,   -168,    462,    -108,     6;
   -2380,    763,   3836,   -1624,   196,    -7;
  -14208,  21248,  29560,  -21472,  4256,  -320,    8;
  -73836, 302571, 199998, -269127, 78840, -9387,  486,   -9;

Mathematica

P[x_, n_]:= P[x, n]= If[n<2, (-x)^n, ((2*n-x)*P[x, n-1] -n*P[x, n-2])/(n+1)];
Table[CoefficientList[(n+2)!*P[x, n]/2, x], {n, 0, 12}]//Flatten

Prog

(Magma)
function P(n, x)
  if n eq 0 then return (-x)^n;
  else return ((2*n-x)*P(n-1, x) - n*P(n-2, x))/(n+1);
  end if;
end function;
q:= func< n, x | Factorial(n+2)*P(n, x)/2 >;
R<x>:=PowerSeriesRing(Rationals(), 30);
A136532:= func< n, k | Coefficient(R!( q(n, x) ), k) >;
[A136532(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 25 2023
(SageMath)
def P(n, x):
    if (n<2): return (-x)^n
    else: return ((2*n-x)*P(n-1, x) - n*P(n-2, x))/(n+1)
def q(n, x): return factorial(n+2)*P(n, x)/2
def A136532(n, k):
    P.<x> = PowerSeriesRing(QQ)
    return P( q(n, x) ).list()[k]
flatten([[A136532(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 25 2023

Crossrefs

Cf. A021009.

Sequence in context: A362273 A094357 A340003 * A368041 A030417 A275239

Adjacent sequences: A136529 A136530 A136531 * A136533 A136534 A136535

Keyword

tabl,sign

Author

Roger L. Bagula, Mar 23 2008

Extensions

Edited by G. C. Greubel, Jul 25 2023

Status

approved