Real-rooted atlas

A094638 → θ+1 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21318637
37015010524349
4446131014256801203981
53276123201837513580497072053241
627252125552239505242060136640408245040816873
7253296138594432343364171860321114514743683737164032014144985
826022241652623245740352720463247061890544096304171242123779712362880272897145
9292881602121023526805076521269347464151671019512036472146342533122139318724189838436288005805315405
10358457760291888907210667119452230362574643255065544531440641694210193372629601859872286761578450545440039916800135006205005
114740577920429127395841761671031124328861748647073564054708069563522026557202822173794654154721532344053244111409016065949998404790016003407627547765
126737553216067163948467230626016529688446193923024156889258940702067636481739419821752599649139257857001127114830378084257760857640062844098803292561564160622702080092780308131525
131024208720640111547166215685597339309265617133181024044835689561376612053421635139100859109606945889548945560985349630327224115731413885629219748045657976474968102082977285121390753653120871782912002710620512647425
1416583626886400195998593501440107384994544276836148138246696328351740885699640140090915821359361759197478126988916799371259144760122580094725522726800412578742640282325654696571285044628845446417568715367923222275960215040130767436800084584820173727825
Technical details
Node id
n-d671506ece7c3beb · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-179acb9fd453bef3

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:        3,         2;
  2:       13,        18,         6;
  3:       70,       150,       105,         24;
  4:      446,      1310,      1425,        680,        120;
  5:     3276,     12320,     18375,      13580,       4970,        720;
  6:    27252,    125552,    239505,     242060,     136640,      40824,      5040;
  7:   253296,   1385944,   3234336,    4171860,    3211145,    1474368,    373716,     40320;
  8:  2602224,  16526232,  45740352,   72046324,   70618905,   44096304,  17124212,   3779712,   362880;
  9: 29288160, 212102352, 680507652, 1269347464, 1516710195, 1203647214, 634253312, 213931872, 41898384, 3628800;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(2*n^3-2*n^2)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (2*n^2 - 2*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] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (2*n^3 - 2*n^2)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 1 + (2*n^2 - 2*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

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.