Real-rooted atlas

A078812 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
261218
335144144323
42041272259217285796
511899936306724147220736104005
66930726123006726151686220802488321866294
74039150889626447047257600110937608957952298598433489287
82354163465456216743047466688015303168018662400012541132835831808600940872
9137210523108832168986592700835328180309888029471662082988969984171992678442998169610783446409
1079972141516527961269551232615818419219075585536391335598085317440307246151368704232190115845159780352193501094490
1146611179982774512926815435251482884608186548624640460761219072785660166144913281122304692700512256309586821120619173642243472236254411
122716698606304328424661326690244139569756801718010777600496401725030410189921222656148614573588481509235752960010164767293440408654603878474300837068862306751484908
13158340798140103234352463179800880322585099776015088999749120499394716139521199220741734402105345177026562681989329715202417873072947201464345663897605349660268953689161004482561118049290473933
14922877802625331597346031941849484802449930662144012752619969024047574080942284813086161831854082687410538938368412051545273139246589462721331203775968539836416207745140444364869545583496396810699320537907220062580477045886
1553789260175159053794272021741721571520182100208112640104405987784960043348197812060161344066473801318431617322914152448567038202506772487715807575828070478416731938226176577228343020093442909323576265932889874292518420481283918464548864360008399296352015
Technical details
Node id
n-251095b7f718f862 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-cb87bae463f34fa7

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:       6,        12;
   3:      35,       144,        144;
   4:     204,      1272,       2592,       1728;
   5:    1189,      9936,      30672,      41472,       20736;
   6:    6930,     72612,     300672,     615168,      622080,      248832;
   7:   40391,    508896,    2644704,    7257600,    11093760,     8957952,     2985984;
   8:  235416,   3465456,   21674304,   74666880,   153031680,   186624000,   125411328,    35831808;
   9: 1372105,  23108832,  168986592,  700835328,  1803098880,  2947166208,  2988969984,  1719926784,   429981696;
  10: 7997214, 151652796, 1269551232, 6158184192, 19075585536, 39133559808, 53174403072, 46151368704, 23219011584, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 12*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] = 6 + 12*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (6 + 12*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