Real-rooted atlas

A103327 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-14 - 10\,x + 20\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-12 - 20\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
24432480
33283761208832
4244838402016384168704
5182723638426688864011203291136
613638432921631046414284832320307264954368
7101798428875523329664199104064960011020880641289994240
8759833624764416337664002487500810647040263782435123220480256104660992
9567147522088145923286978562879139841528309765008896098488321062912506885121096024064
104233246721737531392310126694431494144001997649920816807936214050816344555523087360122880102411477712896
1131597383681430457344028548093952329838858242434942976011950403584392820736084844953611441971286732802928642048120196169728
1223584608256116738818048257604427776333705543680281139998720161154859008638460067841738296524831649587203640524802369945668812840961258710630400
13176037912576945752776704228652277760032825234718723108480983040203926627123294667371315231182667776071893426176112284057601117569024632586241597440819213181388849152
1413139648675847614541725696200165266391043154366601625633173319696384245194925015041306127189606450563889233921415346536448281093210112381871226883327787008165494784367001616384138037296103424

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-4 - 8\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
Technical details
Node id
n-2b6c5dd961f03a94 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-c00c60332ef216a6)
Fingerprint
f6-935c57c7e92571a4

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:        6,         2;
  2:       44,        32,         4;
  3:      328,       376,       120,         8;
  4:     2448,      3840,      2016,       384,        16;
  5:    18272,     36384,     26688,      8640,      1120,       32;
  6:   136384,    329216,    310464,    142848,     32320,     3072,      64;
  7:  1017984,   2887552,   3329664,   1991040,    649600,   110208,    8064,     128;
  8:  7598336,  24764416,  33766400,  24875008,  10647040,  2637824,  351232,   20480,   256;
  9: 56714752, 208814592, 328697856, 287913984, 152830976, 50088960, 9848832, 1062912, 50688, 512;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (20*n-14+(12*n-10)*x)*P(n-1,x) - (12+20*x+8*x^2)*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[((20*n - 14 + (12*n - 10)*x)*P[n - 1, x] - (12 + 20*x + 8*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((20*n - 14 + (12*n - 10)*x)*v[i - 1] - (12 + 20*x + 8*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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