Real-rooted atlas

A046716 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(32 + 48\,x + 18\,x^{2} - 16\,n - 24\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 3\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
32933971
421735418027778
522974950371710808112125
6343499491798991466838829243284012
76749652319567318855621937237576741066777299241891
81627648167383420116153001106715853551252791518920117496002187378595022
94615277932241102780466092279053817880683726030699154680872435654077835219448656118409947641
101499313877383631891159203986810890284158462230247262918139137622938949478465488549499894974824776749196831029827400400
115482586875013456292897101967978114778715806835148518165859841212651159694012033154031356284491617516244350282324547359218899170095904964998958518719
122227273873386515664626116611849558889114461892879130078557111418759345404799622627492307405629015114542182257024873083575936451222424592488162115526481352871721771474565303338264082
13994870668959209772443754816888227252740651158755576723356137312928133797160258172280273770406868006565690765431836792846466855363243610022526806028822235179402035185833098693401575921165761645568531441353016345110857429
1448451779617935997411704515243112121160297567279113435337815153463165295616019877909570716494681276654657988977656203067397485525133405446528747743341150412232838227517047227500765126943210006280593227522912844740646264329755466070310613159432329793105387299603252
15255481833907883635723585049454076302107100466208550567427526261468892913316428313463856047929818346105592340510705583206304560165352156595779255397201101129828164745211179482805255099561831462454876887408242356388960656257445545519577637762640335372114841270060122925318076558110947829692724646021507044539675
Technical details
Node id
n-9a60224d294916c0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-40f5d787e72f16e0

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:           5,           3;
   3:          29,          33,            9;
   4:         217,         354,          180,           27;
   5:        2297,        4950,         3717,         1080,           81;
   6:       34349,       94917,        98991,        46683,         8829,          243;
   7:      674965,     2319567,      3188556,      2193723,       757674,       106677,         729;
   8:    16276481,    67383420,    116153001,    106715853,     55125279,     15189201,     1749600,       2187;
   9:   461527793,  2241102780,   4660922790,   5381788068,   3726030699,   1546808724,   356540778,   35219448,      6561;
  10: 14993138773, 83631891159, 203986810890, 284158462230, 247262918139, 137622938949, 47846548854, 9499894974, 824776749, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n-32+(24*n-48)*x+(9*n-18)*x^2)*P(n-2,x) + (4*n-3+(3*n-3)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(16*n - 32 + (24*n - 48)*x + (9*n - 18)*x^2)*P[n - 2, x] + (4*n - 3 + (3*n - 3)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = -(16*n - 32 + (24*n - 48)*x + (9*n - 18)*x^2)*v[i - 2] + (4*n - 3 + (3*n - 3)*x)*v[i - 1]); 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