Real-rooted atlas

A207824 → x ↦ x+1 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1/4, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-6\,x - 32\,x^{2} + 2\,n + 18\,n x + 32\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 12\,x^{2} - 32\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1167
21123548
3118106204329
412421382811892255
513035620886051693015456
61365354200183664240840391105937
7142750738043210150516288788235416726103
8148100111844869554055761173518192573613721054976784
91541288178081572699199263563764881632812637733799721434111385
101601611254882631161851696899636529793336643621838189787646611179233802911
11166197035100414756341258419949272828516002395584354592546465253607022716698601602508992
12172236546860623745587563240161660200695536727671670186681192032165263953341936196158340798110983760033
1317827966098490293595830027500069643825291219118956146149990340141778394802218183992021109664455922877802675283811239
141843263776881266474149537521318749638823814564501251426174206684885035010985010538206488015098417758013254482856053789260175516002918640
Technical details
Node id
n-f2ffa8b9a586a0aa · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-684f9d5a83927374

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  6;
  2: 1, 12,   35;
  3: 1, 18,  106,   204;
  4: 1, 24,  213,   828,   1189;
  5: 1, 30,  356,  2088,   6051,   6930;
  6: 1, 36,  535,  4200,  18366,  42408,   40391;
  7: 1, 42,  750,  7380,  43210, 150516,  288788,  235416;
  8: 1, 48, 1001, 11844,  86955, 405576, 1173518, 1925736,  1372105;
  9: 1, 54, 1288, 17808, 157269, 919926, 3563764, 8816328, 12637733, 7997214;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (2*n+(18*n-6)*x+(32*n-32)*x^2)*P(n-1,x) - (x+12*x^2+32*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (18*n - 6)*x + (32*n - 32)*x^2)*P[n - 1, x] - (x + 12*x^2 + 32*x^3)*D[P[n - 1, x], x])/(2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (18*n - 6)*x + (32*n - 32)*x^2)*v[i - 1] - (x + 12*x^2 + 32*x^3)*deriv(v[i - 1], x))/(2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023