Real-rooted atlas

A114687 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(9 + 24\,x + 16\,x^{2} - 9\,n - 24\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(5 + 4\,x + 10\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
229441689
318540828864945
4125736243776166425610577
5892531740436802880089601024123129
6654452766604730404156801958404608040961476841
7491825240856049212805407360342272012328962293761638418130401
8376820920974736498480646576281652347904255180807340032111411265536226739489
9293244051828306444954469767630371847324047364513259521768488964187750453084162621442878666857
10231153133159563174048557373608551843200961454592071722721283569664000115851264023101440024903680104857637006326777
1118418010651394393255247080494560933350985601203173452801054513152006376469299226401013760725581824012399411201153433604194304480750990993
1214810069497122012836152452629096128997539889280145064582784014614092472321041191534592525513719808185007144960438016409606505365504528482304167772166301611631473
1312002965780510689891309244321876165984104823479537921697715054464019333301425152158690373304329462056288256407343366144012400939499522564082892803347893452823991418886710886483240669582937
14979470140661937691379853241035542479568108622935849792193908395769088246350218290176229009996959744157656031543296803795284131843001112828313680075858903041462222454784169466658816108045271042684354561106980509493641
Technical details
Node id
n-31423e7dce7f16c2 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-284ad1b575773d67

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,         4;
  2:       29,        44,        16;
  3:      185,       408,       288,        64;
  4:     1257,      3624,      3776,      1664,       256;
  5:     8925,     31740,     43680,     28800,      8960,      1024;
  6:    65445,    276660,    473040,    415680,    195840,     46080,      4096;
  7:   491825,   2408560,   4921280,   5407360,   3422720,   1232896,    229376,    16384;
  8:  3768209,  20974736,  49848064,  65762816,  52347904,  25518080,   7340032,  1114112,   65536;
  9: 29324405, 182830644, 495446976, 763037184, 732404736, 451325952, 176848896, 41877504, 5308416, 262144;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(9*n-9+(24*n-24)*x+(16*n-16)*x^2)*P(n-2,x) + (10*n+5+(8*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n - 9 + (24*n - 24)*x + (16*n - 16)*x^2)*P[n - 2, x] + (10*n + 5 + (8*n + 4)*x)*P[n - 1, x])/(n + 2)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n - 9 + (24*n - 24)*x + (16*n - 16)*x^2)*v[i - 2] + (10*n + 5 + (8*n + 4)*x)*v[i - 1])/(n + 2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023