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
Examples
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;
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.
Programs
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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
-
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 *) -
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 -
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
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)
Extensions
Edited by N. J. A. Sloane, Oct 01 2006
Name corrected and more terms from Michel Marcus, Feb 07 2014