Real-rooted atlas

A172431 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-7 - 37\,x - 71\,x^{2} - 59\,x^{3} - 18\,x^{4} + 13\,n + 59\,n x + 97\,n x^{2} + 69\,n x^{3} + 18\,n x^{4}\right)\,P_{n-1}(x) + \left(-5 - 34\,x - 90\,x^{2} - 116\,x^{3} - 73\,x^{4} - 18\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
28262155
32110116185368
4553508318723412449
51441140359456384399136516280
63773570140312928834234212385461108199
798710879512131334452078151933719951721845719072
8258432494178215556708108316013438621038057456324873814778785
967659557559843821793865086573788897581292205365860205831534952531758632
101771127772019546628130804221337974119645653086390467596802693934091653301398101211059991
1146368799080624478829210284908575421973055423052100383362723502585777851321464654038959355924051402652240
121213932280516195933781017944603561483858839494761595750422211097617620307714851385366910636084323176471456223696219321678001
13317811646396160556134345964514134473962537550135077749136272119653459681382268799811798411150723180092630140993467655381518947848561949553848
14832040182148501847871931151384176492234170815273017790354663344856261194349684437115957865563158866638682062536939026280140945184463300761902357913941411701328775
Technical details
Node id
n-a185b3495c199876 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-014a0c4e84d47b3f

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:    3,     5;
  2:    8,    26,     21;
  3:   21,   101,    161,      85;
  4:   55,   350,    831,     872,     341;
  5:  144,  1140,   3594,    5638,    4399,    1365;
  6:  377,  3570,  14031,   29288,   34234,   21238,    5461;
  7:  987, 10879,  51213,  133445,  207815,  193371,   99517,   21845;
  8: 2584, 32494, 178215,  556708, 1083160, 1343862, 1038057,  456324,   87381;
  9: 6765, 95575, 598438, 2179386, 5086573, 7888975, 8129220, 5365860, 2058315, 349525;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = (13*n-7+(59*n-37)*x+(97*n-71)*x^2+(69*n-59)*x^3+(18*n-18)*x^4)*P(n-1,x) - (5+34*x+90*x^2+116*x^3+73*x^4+18*x^5)*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[((13*n - 7 + (59*n - 37)*x + (97*n - 71)*x^2 + (69*n - 59)*x^3 + (18*n - 18)*x^4)*P[n - 1, x] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*x^2)]];
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] = ((13*n - 7 + (59*n - 37)*x + (97*n - 71)*x^2 + (69*n - 59)*x^3 + (18*n - 18)*x^4)*v[i - 1] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*n*x^2)); 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

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