A122766 Triangle read by rows: let p(n, x) = x*p(n-1, x) - p(n-2, x), then T(n, x) = d^2/dx^2 (p(n, x)).

2, -2, 6, -6, -6, 12, 6, -24, -12, 20, 12, 24, -60, -20, 30, -12, 60, 60, -120, -30, 42, -20, -60, 180, 120, -210, -42, 56, 20, -120, -180, 420, 210, -336, -56, 72, 30, 120, -420, -420, 840, 336, -504, -72, 90, -30, 210, 420, -1120, -840, 1512, 504, -720, -90, 110

Offset

1,1

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Formula

From G. C. Greubel, Dec 31 2022: (Start)

T(n, k) = 2*(-1)^binomial(n-k+1, 2)*binomial(k+1,2)*binomial(floor((n+k +2)/2), k+1).

T(n, 1) = 2*(-1)^binomial(n,2)*binomial(floor((n+3)/2), 2)

T(n, n) = 2*A000217(n).

Sum_{k=1..n} T(n, k) = 2*A104555(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = 2*([n=1] - [n=2]). (End)

Example

Triangle begins as:
    2;
   -2,    6;
   -6,    6,   12;
    6,  -24,  -12,   20;
   12,   24,  -60,  -20,   30;
   12,   60,   60, -120,  -30,   42;
  -20,  -60,  180,  120, -210,  -42,  56;
   20, -120, -180,  420,  210, -336, -56,  72;

Mathematica

(* First program *)
p[0, x]=1; p[1, x]=x-1; p[k_, x_]:= p[k, x]= x*p[k-1, x] -p[k-2, x]; b = Table[Expand[p[n, x]], {n, 0, 15}]; Table[CoefficientList[D[b[[n]], {x, 2}], x], {n, 2, 14}]//Flatten
(* Second program *)
T[n_, k_]:= 2*(-1)^Binomial[n-k+1, 2]*Binomial[k+1, 2]*Binomial[Floor[(n +k+2)/2], k+1]; Table[T[n, k], {n, 14}, {k, n}]//Flatten (* G. C. Greubel, Dec 31 2022 *)

Prog

(PARI) tpol(n) = if (n <= 0, 1, if (n == 1, x -1, x*tpol(n-1) - tpol(n-2)));
lista(nn) = {for(n=0, nn, pol = deriv(deriv(tpol(n))); for (k=0, poldegree(pol), print1(polcoeff(pol, k), ", "); ); ); } \\ Michel Marcus, Feb 07 2014
(Magma)
A122766:= func< n, k | 2*(-1)^Binomial(n-k+1, 2)*Binomial(k+1, 2)*Binomial(Floor((n+k+2)/2), k+1) >;
[A122766(n, k): k in [1..n], n in [1..14]]; // G. C. Greubel, Dec 31 2022
(SageMath)
def A122766(n, k): return 2*(-1)^binomial(n-k+1, 2)*binomial(k+1, 2)*binomial(((n+k+2)//2), k+1)
flatten([[A122766(n, k) for k in range(1, n+1)] for n in range(1, 15)]) # G. C. Greubel, Dec 31 2022

Crossrefs

Cf. A000217, A046854, A066170, A122765, A130777.

Sequence in context: A300951 A077081 A084700 * A291185 A320140 A033742

Adjacent sequences: A122763 A122764 A122765 * A122767 A122768 A122769

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Sep 22 2006

Extensions

Edited by N. J. A. Sloane, Oct 01 2006

Name corrected and more terms from Michel Marcus, Feb 07 2014

Status

approved