Real-rooted atlas

A091044 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 96\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1122436
21284804321040
31344729612672691228224
41408099840255744276480103680749824
51474561290240436838470778885391360149299219768320
6154419216072704676823041464238081695375369754214420901888519704576
716171008195133440982941696266305536041538355203678732288167215104028665446413648674816
8169345024232429977613630242816443925135368724381696010491553382474243506176275188285443869835264358307921920
91773404160272813260801825576058886947654860801648687841280251173011456024308298547201417219670016438581329920515978035209405024436224
101857132953631649169408023798407495681036508567961628838288424960532626926469126562858401792052671037833216258814582456326810910064640681091006464246854051692544
11194481487872363686422118430358590455808148926917246976475456730038272103276328937062415484172139233281589414153158656108157251826483245587691365990410352583298252889161004482566479059605258240
1220366365818884146321162240038045290070016020758482911232007480568866406400186908698843545603305133614039040041422074559856640362037708324864002125003940167680077916811139481601545457411031040115909305827328170051508556005376
132132793517670446957776705945646976701405593602822008530272256011331125662187520032021055660883968065281119058132992096750036943110144010357140416259686407838903423729664004023340793384140801298184225266073602271822394215628814979048753070084463215049070084096
142233490368102405287875151134720572809177013944323756773534072832001663261371216691200524588615498465280012118440841681305600207503062813428940802633340547057582080024483479740533964800162633634887499776007381104595084247040211604028718369996832953907256754176019258776968232960117142555576185651200
Technical details
Node id
n-b4a209047dc924c9 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-53dcb76f02d19ad5

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:         12,          24;
  2:        128,         480,          432;
  3:       1344,        7296,        12672,         6912;
  4:      14080,       99840,       255744,       276480,        103680;
  5:     147456,     1290240,      4368384,      7077888,       5391360,       1492992;
  6:    1544192,    16072704,     67682304,    146423808,     169537536,      97542144,      20901888;
  7:   16171008,   195133440,    982941696,   2663055360,    4153835520,    3678732288,    1672151040,     286654464;
  8:  169345024,  2324299776,  13630242816,  44392513536,   87243816960,  104915533824,   74243506176,   27518828544,   3869835264;
  9: 1773404160, 27281326080, 182557605888, 694765486080, 1648687841280, 2511730114560, 2430829854720, 1417219670016, 438581329920, 51597803520;
%F Formula
The row polynomials satisfy P(n,x) = -(16+96*x+144*x^2)*P(n-2,x) + (12+24*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 12 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12 + 24*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 96*x + 144*x^2)*P[n - 2, x] + (12 + 24*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] = 12 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 + 96*x + 144*x^2)*v[i - 2] + (12 + 24*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

Same fingerprint ?

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