Real-rooted atlas

A109692 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} - 16\,n - 44\,n x - 30\,n x^{2} + 20\,n^{2} + 48\,n^{2} x + 28\,n^{2} x^{2} - 8\,n^{3} - 16\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2 + 6\,x - 12\,n - 16\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2820937
34820020460512
43842272373223845259297
53840296326799268624320705670207828
646080440064130483218207041328420491364727655504229
764512073589762684889648106944481843202760244884710641081080168298848
810321920137097216595159296130668902416729543201308956928618579864162515520182432255830517313
918579456028186951681420884633637010679808578803195205768949283237095416624149434024323436487370344594250225613728900
10371589120063434096640364509739008110017103667220366100358402473461141696202069209025611043685239043881449743847943995170072020198259641749501125
118174960640015516175564801001738830233634410035499008736849105011201057387180204801050550239571847279271769203234629895813680108043950063601993507320420164991726900450924951002400
12196199055360040999303249920293948124954624113304075991449627577263726233604571432762322432535864284185529645084715686261762714629069809648114429170650224032119536935098853978712394320411104386192522904429626019025
13510117543936001164081412177920918012512069222439257330589597696107108409435535360201589042999084032272658169455975296270324341980146176197381118682662048105157947419326240398191015319372881016332023182472015688106010895501106819501287501255533493164570900
1414283291230208003534891427823616030418880057258803214296897608536555524323970371593912320911480566968569856013984483599238455296159825336615141877761373777321076147744088658560882209961604238125881248233392145742650252269955234120381361138046048715279773747300320187070015312573868751753698441925
Technical details
Node id
n-b2364ce52abc9924 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-df56de28b5551755

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:         2,          2;
  2:         8,         20,           9;
  3:        48,        200,         204,          60;
  4:       384,       2272,        3732,        2384,         525;
  5:      3840,      29632,       67992,       68624,       32070,        5670;
  6:     46080,     440064,     1304832,     1820704,     1328420,      491364,       72765;
  7:    645120,    7358976,    26848896,    48106944,    48184320,    27602448,     8471064,     1081080;
  8:  10321920,  137097216,   595159296,  1306689024,  1672954320,  1308956928,   618579864,   162515520,   18243225;
  9: 185794560, 2818695168, 14208846336, 37010679808, 57880319520, 57689492832, 37095416624, 14943402432, 3436487370, 344594250;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(16*n^3-48*n^2+44*n-12)*x+(8*n^3-28*n^2+30*n-9)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(8*n^2-16*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x])/(2*n - 3)]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1])/(2*n - 3)); 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. [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.