Real-rooted atlas

A123519 → basis A013609

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 11 + 28\,x + 16\,x^{2}\)
\(P_{1}(x) = 41 + 152\,x + 176\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
011281655
14115217664433
215374412969602563409
35713436798489604864102426839
4213115284443846681655040235524096211303
5795366224230688435840482560313344110592163841663585
629681281424114326425992963618560316006416916485079046553613097377
7110771117819654680641453440024388864268011521929625687818242293760262144103115431
84134034874444254417767741056015207449620160000018269798411180441644236800102236161048576811826071
91542841199731921157968163968749448935513601387896832151789568011692277766218711042175795204508876841943046391493137
1057579618118968851763825619732764805009630720892419276811440472064106366894087120158720334757888010496245761971322881677721650320119025
11214890033278174042279514960956678400027038077440543473971207994076364887139418112704055214084165337088017543725056498283315285563801667108864396169459063
12801980511316035940991169120045411919360141409815040316617705472525712171008657684037632623059599360444887990272235738759168899091005442333710745636909875202684354563119035553479
13299303201525711856042631393600211727658240720248860160177776268492832893787422724646445416448505256902656042301369876482704963272704129760860569645212919398410804527104015837691904107374182424556114968769
141117014753209096511041816423574409720242880003586496258560967542847488019742853537792310856125972483817299817267236685936066560275046622822401592037841305669756140912642237006872576495263416320676457349124294967296193329884196673

Computed in your browser ?

Technical details
Node id
n-d5dd236ae5fa65ba · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-84818a0aba74ba02

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

OEIS matches ?

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      11,       28,        16;
  1:      41,      152,       176,        64;
  2:     153,      744,      1296,       960,       256;
  3:     571,     3436,      7984,      8960,      4864,       1024;
  4:    2131,    15284,     44384,     66816,     55040,      23552,       4096;
  5:    7953,    66224,    230688,    435840,    482560,     313344,     110592,      16384;
  6:   29681,   281424,   1143264,   2599296,   3618560,    3160064,    1691648,     507904,     65536;
  7:  110771,  1178196,   5468064,  14534400,  24388864,   26801152,   19296256,    8781824,   2293760,    262144;
  8:  413403,  4874444,  25441776,  77410560, 152074496,  201600000,  182697984,  111804416,  44236800,  10223616,  1048576;
  9: 1542841, 19973192, 115796816, 396874944, 893551360, 1387896832, 1517895680, 1169227776, 621871104, 217579520, 45088768, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 28*x + 16*x^2, P(1,x) = 41 + 152*x + 176*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 11 + 28*x + 16*x^2;
P[1, x_] := 41 + 152*x + 176*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 4*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] = 11 + 28*x + 16*x^2; v[2] = 41 + 152*x + 176*x^2 + 64*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 4*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-125a67b606fd2347
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-125a67b606fd2347
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-125a67b606fd2347
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-125a67b606fd2347

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023