Real-rooted atlas

A094638 → D → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2} - n^{3} - 4\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 4\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
27815
34611672234
4326131617527684162
52556144163025227968960084792
6222121622564712526798084867201382401960488
72129761919984717739214235168157904009282816225792050876656
8223934424095088110676144281234208426816960386721024193600512412876801466670960
92565936032156726417573288165468569536105959392801308645657610057146624439551590483607552046544258880
10318540960456361430428977634896107016366656253269570480398253565056415979266304278228312064108095938560185794560001613282265280
11426157632068779173888498330890784213420702348859821849288001146535944134415213592698752137976470118408183301986304286529863680044962283520060662586203520
1261148511360109868492390489543518401604369127981043214179461477904032133659868691251880676944140859665652517683247891939514163225547026857984081488980869120117719433216002460050561082240
1393703042944018561311525376168209637794496922095447804928340509120074560089227043627228161701050493234937623769163930163200241570748934051841741203295309824084472954135511042476131027517440331576403558400107041378544665600
141527595251840033089337692928033027196509832322011125819473510483349708684356480248251629785894400546632583167519296900873616787628032111108478720121856010127339354246348806630000461586923522951684434994135048008295388807168099983038611456004974936155633740800
Technical details
Node id
n-bc4bccd8525de23a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-23394a58ed9a04a5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        1;
  2:        7,         8;
  3:       46,       116,         72;
  4:      326,      1316,       1752,        768;
  5:     2556,     14416,      30252,      27968,        9600;
  6:    22212,    162256,     471252,     679808,      486720,      138240;
  7:   212976,   1919984,    7177392,   14235168,    15790400,     9282816,     2257920;
  8:  2239344,  24095088,  110676144,  281234208,   426816960,   386721024,   193600512,   41287680;
  9: 25659360, 321567264, 1757328816, 5468569536, 10595939280, 13086456576, 10057146624, 4395515904, 836075520;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(n - 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.