Real-rooted atlas

A046716 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 20\,x + 66\,x^{2} + 80\,x^{3} + 32\,x^{4} - n - 10\,n x - 33\,n x^{2} - 40\,n x^{3} - 16\,n x^{4}\right)\,P_{n-2}(x) + \left(-3\,x - 6\,x^{2} - 3\,x^{3} + n + 6\,n x + 9\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2257
35252959
416123297217653
5656922649423822979941
6326451324278628137682834349203107
7195733681236646864131171519617302236749655256799
81370028347524763291180031232974687536585824655194716276481164035513
91096012658016278797941648483005992804191366146700189598153414511195644615277935969551721
10986410274907973374379002390863026107562037143178475394361479918480746846774465130635779814993138773247761828287
119864101311028161438086173536260713014195066333213711454979979177773123840129959122382623244786011423202629397790954825868750111540465933699
12108505112382240170760835139705576965144463361970694096615754562833623484696768653871051980517034491574288581284061541269098460138835386490639722272738733865595866101149093
1313020613455071862202090085846077996423440913946921977823531035080122935145612855846215915073427977378472326658906606808699488825529195937287945391772608847421401047934162699487066895920933770070139476541
1416926797486722784648329141823001961301692406594278231369615648154822793254152931870633801591970830342107187326579388279252635388793024914436415523257381534535928019070243478441760300072796667218168619790055840484517796179359972083834869000200347
15236975164805110118407154972366765024567363117411776107457281009128473022031832913343955543248915851049274358773283269478757393865918963933938321933119432694656305133460305170037728806421333444562770023757160126349034503749608622121824072930274068912554818339078836357139060837772684549159
Technical details
Node id
n-ad2dfc31d01b8299 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b761c8caa7513a1c

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,        5;
   3:      5,       25,        29;
   4:     16,      123,       297,        217;
   5:     65,      692,      2649,       4238,        2297;
   6:    326,     4513,     24278,      62813,       76828,       34349;
   7:   1957,    33681,    236646,     864131,     1715196,     1730223,      674965;
   8:  13700,   283475,   2476329,   11800312,    32974687,    53658582,    46551947,    16276481;
   9: 109601,  2658016,  27879794,  164848300,   599280419,  1366146700,  1895981534,  1451119564,   461527793;
  10: 986410, 27490797, 337437900, 2390863026, 10756203714, 31784753943, 61479918480, 74684677446, 51306357798, 14993138773;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-2+(10*n-20)*x+(33*n-66)*x^2+(40*n-80)*x^3+(16*n-32)*x^4)*P(n-2,x) + (n+(6*n-3)*x+(9*n-6)*x^2+(4*n-3)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (10*n - 20)*x + (33*n - 66)*x^2 + (40*n - 80)*x^3 + (16*n - 32)*x^4)*P[n - 2, x] + (n + (6*n - 3)*x + (9*n - 6)*x^2 + (4*n - 3)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 5*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (10*n - 20)*x + (33*n - 66)*x^2 + (40*n - 80)*x^3 + (16*n - 32)*x^4)*v[i - 2] + (n + (6*n - 3)*x + (9*n - 6)*x^2 + (4*n - 3)*x^3)*v[i - 1])/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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