Real-rooted atlas

A116414 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-8 - 12\,x - 4\,x^{2} + 18\,n + 20\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-13 - 29\,x - 20\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
222341369
39521616240513
4409122613656681213789
51760654896627060255136427945
675733364861824600763252992941093206037
7325851683283701584488803239501390083280432801518993
81402068255982114023307291627717551587556563422113176984111198493
9603275398832011654310197481002137483515317436726302421958803838172952482558521
102595757190366706250855512097806015277174513147457478053997315487688303760128431488573608644773
111116896089981768327986780713789400103019493010350142327383500883738417361315980693065749642515262657204487100705
124805752942190012816903280164085798600663442299076218625086350482698386521502417049914105313579501110047311395032479716133080169069
132067807651964771692858188276422811070880411580278205321942402450724143076360713374801913292225974736692802088961694395491740454382032391484243876313161
1488973123890969359624302194318512471782117624754537800235579971013638181251936431068913092019259224874790480368054317033494678031352268139005035514707629471744531797924789621
Technical details
Node id
n-837390f0e0e9eb6f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9e17c894e280f681

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:      5,       4;
  2:     22,      34,       13;
  3:     95,     216,      162,       40;
  4:    409,    1226,     1365,      668,      121;
  5:   1760,    6548,     9662,     7060,     2551,      364;
  6:   7573,   33648,    61824,    60076,    32529,     9294,    1093;
  7:  32585,  168328,   370158,   448880,   323950,   139008,   32804,    3280;
  8: 140206,  825598,  2114023,  3072916,  2771755,  1587556,  563422,  113176,   9841;
  9: 603275, 3988320, 11654310, 19748100, 21374835, 15317436, 7263024, 2195880, 383817, 29524;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (18*n-8+(20*n-12)*x+(4*n-4)*x^2)*P(n-1,x) - (13+29*x+20*x^2+4*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[((18*n - 8 + (20*n - 12)*x + (4*n - 4)*x^2)*P[n - 1, x] - (13 + 29*x + 20*x^2 + 4*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] = ((18*n - 8 + (20*n - 12)*x + (4*n - 4)*x^2)*v[i - 1] - (13 + 29*x + 20*x^2 + 4*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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023