Real-rooted atlas

A207606 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 89 + 216\,x + 128\,x^{2}\)
\(P_{1}(x) = 610 + 2208\,x + 2624\,x^{2} + 1024\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
089216128433
16102208262410246466
24181201203590428160819296557
328657172080409664483328282624655361441889
41964181413696420838466324485836800271974452428821531778
513462691129513640358592796108809363456065667072254279684194304321534781
692274658842240836866284887351244812864921601206026240702808064232783872335544324801489937
763245986681481440324766060889842790401589991014418668453888145424384007247757312209715200026843545671700814274
843349443751879155602781681292887997725696181887111168256672432128250441891840166841024512726285680641865626419221474836481070710724173
9297121507339101882976232973354368829534304256196129167360032331354603523791930261504316417454899218410310205447113539584001642824990721717986918415988960048321
10203650110742922949858561915811731456758252923904020183429038080380656091791365215815362150452317822910464381279849676801968906960896068386616770561434519076864137438953472238763690000642
1113958386244521698831066081551606865267267574664220672199982945505280424693561098240665840018522112780325814796288683597447036928442136013045760205218906112000647337470853121243822528921610995116277763565466389961309
12956722026041160135576599601240557336901125895686695270401920294503264256453465288255078480062704648192001073668302884044810989660662661120854404259800678449667817854730242093452959416320604800114753536107202383708160879609302220853243232159418993
13655747031984211757879672144098098252845779250518518919895041795862793356083246680232642871296916552762956185601384266191056404481621374914223144961472834474742906881029145943762534405432369125261312020968786253316096558771809235763291919172082073670368744177664795083016001323586
144494557021285385949778194885676834523392859524262125480206182416420491616757760046589599908603494410070225247474810881691491861075591168223138573213866393623195400211005440001893702958642626560120148914081890304058076623367936409620675711428932403251127290691584000784611497580953656294995342131211873002007860434797
Technical details
Node id
n-4114002a83b4026e · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4959924f34dfec13

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         89,         216,          128;
  1:        610,        2208,         2624,         1024;
  2:       4181,       20120,        35904,        28160,          8192;
  3:      28657,      172080,       409664,       483328,        282624,         65536;
  4:     196418,     1413696,      4208384,      6632448,       5836800,       2719744,        524288;
  5:    1346269,    11295136,     40358592,     79610880,      93634560,      65667072,      25427968,       4194304;
  6:    9227465,    88422408,    368662848,    873512448,    1286492160,    1206026240,     702808064,     232783872,      33554432;
  7:   63245986,   681481440,   3247660608,   8984279040,   15899910144,   18668453888,   14542438400,    7247757312,    2097152000,    268435456;
  8:  433494437,  5187915560,  27816812928,  87997725696,  181887111168,  256672432128,  250441891840,  166841024512,   72628568064,  18656264192,   2147483648;
  9: 2971215073, 39101882976, 232973354368, 829534304256, 1961291673600, 3233135460352, 3791930261504, 3164174548992, 1841031020544, 711353958400, 164282499072, 17179869184;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 89 + 216*x + 128*x^2, P(1,x) = 610 + 2208*x + 2624*x^2 + 1024*x^3.
%t Mathematica
P[0, x_] := 89 + 216*x + 128*x^2;
P[1, x_] := 610 + 2208*x + 2624*x^2 + 1024*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 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] = 89 + 216*x + 128*x^2; v[2] = 610 + 2208*x + 2624*x^2 + 1024*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 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