Real-rooted atlas

A172431 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(-12 - 8\,x + 40\,n + 24\,n x\right)\,P_{n-1}(x) + \left(-39 - 50\,x - 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
240441296
321734818432781
4115924281884640806191
5616015896162008128200019248576
63268910015212625683616305765824448379561
71733836148369232087592003681601048321612810242960771
89194803703748645291663390083830320145267233600043008230423079456
948759132199003643604968498140803605160017091776528691210229761113605120179854741
102585607112906641228699578037381737631526488017945491269598144180787202990592281600112641401415751
1113710928075046788818494073122705158272260547043217315424008081167362639267845882342484592646983682457610919204736
12727060321432984177611712539488190091657602058775056015649136256854208499233642485769452185601837721602328268817039365324885075858321
13385543872724817611364731084259361303731345281568421545601342247961608391769881638719168512131445250563229557760554977280626196484100096114688662854323131
1420444528200141451734620450750229020876093312640115913771792011030102952967773036741124113251727361638420595204873143808010607387648162843033616512204897484802457605164501096416

Other recurrences

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

\(P_{n}(x) = \left(-9 - 12\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(7 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)
Technical details
Node id
n-d653bfeaea2b665f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-80e18b2692261096)
Fingerprint
f6-03f9938138a9e92b

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:       7,        4;
  2:      40,       44,       12;
  3:     217,      348,      184,       32;
  4:    1159,     2428,     1884,      640,       80;
  5:    6160,    15896,    16200,     8128,     2000,      192;
  6:   32689,   100152,   126256,    83616,    30576,     5824,     448;
  7:  173383,   614836,   923208,   759200,   368160,   104832,   16128,    1024;
  8:  919480,  3703748,  6452916,  6339008,  3830320,  1452672,  336000,   43008,   2304;
  9: 4875913, 21990036, 43604968, 49814080, 36051600, 17091776, 5286912, 1022976, 111360, 5120;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (40*n-12+(24*n-8)*x)*P(n-1,x) - (39+50*x+16*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[((40*n - 12 + (24*n - 8)*x)*P[n - 1, x] - (39 + 50*x + 16*x^2)*D[P[n - 1, x], x])/(4*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] = ((40*n - 12 + (24*n - 8)*x)*v[i - 1] - (39 + 50*x + 16*x^2)*deriv(v[i - 1], x))/(4*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

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