Real-rooted atlas

A142461 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 6\,n\right)\,P_{n}(x) = \left(5 - 110\,x - 55\,x^{2} + 30\,n - 600\,n x - 234\,n x^{2} + 360\,n^{2} x + 540\,n^{2} x^{2} - 216\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 30\,x^{3} + 180\,n x - 36\,n x^{2} - 144\,n x^{3} - 216\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 50\,x - 24\,n - 120\,n x + 72\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 30\,x^{2} + 36\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
2128332
312223334560
41159286583184513440
5111194157926210568279856407680
6178420247281380015844121355235260714909440
71549014357930992344571962930714957158619819215498637378560
8138431844957805645934134816146535747508901202224115682131615372736931160729600
9126902386669303270013720706932859628709287071554451144424011237132417848087200121060737101713840128000
101188316812889867504712990668192032212530042781704436375970952829754205989438598133638406426696025141394158406011104700051456000
111131821780611723638931136263649667938069207352227597023189099409461162705394931103802110731763564155214093211752860539078796437107250197933127032353456128000
121922752477615361347539718127606678906984021138323478511275107597201918769648200199224901681348143462935891692864380489822613202953190166972674600563249409789665536514865613514984653094912000
13164592673578200624884256322254946585922240246161876787113333754578446680018322058128051642842871538841463447346775674729668676700200274830871232375357422666750701103111286160668024995370252884198523782571440830926066810880000
1414521487152042615196393088977502321002912487888173486596854404162051826002338755159865487355256137263387713571258564718227080267456945675789076721277459304333723125859977580381670513079689156511506963008737463664113325177033855673165041006428153484239024367861760000
Technical details
Node id
n-d4ebefdcc50b455a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-05c6a88ef858d065

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: 1,        2;
  2: 1,       28,          3;
  3: 1,      222,        333,            4;
  4: 1,     1592,       8658,         3184,            5;
  5: 1,    11194,     157926,       210568,        27985,            6;
  6: 1,    78420,    2472813,      8001584,      4121355,       235260,            7;
  7: 1,   549014,   35793099,    234457196,    293071495,     71586198,      1921549,           8;
  8: 1,  3843184,  495780564,   5934134816,  14653574750,   8901202224,   1156821316,    15372736,         9;
  9: 1, 26902386, 6693032700, 137207069328, 596287092870, 715544511444, 240112371324, 17848087200, 121060737, 10;
%F Formula
The row polynomials satisfy (6*n-5)*P(n,x) = (30*n+5+(360*n^2-600*n-110)*x-(216*n^3-540*n^2+234*n+55)*x^2)*P(n-2,x) + ((180*n+30)*x-(216*n^2+36*n)*x^2+(216*n^2-144*n-30)*x^3)*d/dx P(n-2,x) - (24*n+10-(72*n^2-120*n+50)*x)*P(n-1,x) + ((36*n-30)*x-(36*n-30)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 5 + (360*n^2 - 600*n - 110)*x - (216*n^3 - 540*n^2 + 234*n + 55)*x^2)*P[n - 2, x] + ((180*n + 30)*x - (216*n^2 + 36*n)*x^2 + (216*n^2 - 144*n - 30)*x^3)*D[P[n - 2, x], x] - (24*n + 10 - (72*n^2 - 120*n + 50)*x)*P[n - 1, x] + ((36*n - 30)*x - (36*n - 30)*x^2)*D[P[n - 1, x], x])/(6*n - 5)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 5 + (360*n^2 - 600*n - 110)*x - (216*n^3 - 540*n^2 + 234*n + 55)*x^2)*v[i - 2] + ((180*n + 30)*x - (216*n^2 + 36*n)*x^2 + (216*n^2 - 144*n - 30)*x^3)*deriv(v[i - 2], x) - (24*n + 10 - (72*n^2 - 120*n + 50)*x)*v[i - 1] + ((36*n - 30)*x - (36*n - 30)*x^2)*deriv(v[i - 1], x))/(6*n - 5)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.