Real-rooted atlas

A172431 → basis A007318 → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-4 - 16\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(5 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15813
2216448133
3853765442561261
4341196841763840128011605
5136597042707236864243206144105469
654614611215934428723228313614336028672953317
7218452135928802241975040259328019783688028161310728596237
887381970896464731212480768204889602093875212902400432537658982477431669
93495254349640237158407418112014623360018759884815566438479822848226099202621440697147165
10139810119262720117854640420817920968611840149659238415646597121085669376473628672115343360115343366275373061
1155924058450744857332496023011545606058316800109548257281383206092812134121472719526297627163361285767168005033164856482551853
122236962136785598427406926881221522304036207612160750234173441110970040321180043837448874950656045733642240151445831682835349504218103808508359743893
138947848515907286161291141382463271719936208534905600487207139328828133834752103367127859294354263244861852876800028069960089682443239424137405399049395240964575304803901
143579139416840133088600765176163210522393601164723187200302945590886458142418001928343482793984896646316032071499880857604139419435008167241999974443969727692865766686720402653184041178011670565
Technical details
Node id
n-247c790d11ff7b06 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-4075b78b60ecafaf

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:      5,       8;
  2:     21,      64,       48;
  3:     85,     376,      544,      256;
  4:    341,    1968,     4176,     3840,      1280;
  5:   1365,    9704,    27072,    36864,     24320,      6144;
  6:   5461,   46112,   159344,   287232,    283136,    143360,     28672;
  7:  21845,  213592,   880224,  1975040,   2593280,   1978368,    802816,   131072;
  8:  87381,  970896,  4647312, 12480768,  20488960,  20938752,  12902400,  4325376,   589824;
  9: 349525, 4349640, 23715840, 74181120, 146233600, 187598848, 155664384, 79822848, 22609920, 2621440;
%F Formula
The row polynomials satisfy P(n,x) = -(4+16*x+16*x^2)*P(n-2,x) + (5+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 16*x + 16*x^2)*P[n - 2, x] + (5 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 16*x + 16*x^2)*v[i - 2] + (5 + 8*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 - 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