Real-rooted atlas

A062190 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(5\,n + n^{2}\right)\,P_{n}(x) = \left(20 - 5\,x + 18\,n + 8\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,x - 5\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
23616153
3165135271328
47158803304011966
530035005286066055111584
61237626208204757280117072167582
75038812994812994863700159251911911391912
820349062016075969647980816660031360294011212264167
98171902877930418608032558401465128385560571204320135113049304
103268760130750402206413020465280113954403907008813960979206120160175093819
11130378955842908511236362512135271580901810344712069401238159885015988584151871431724912
1251895935257690160556794810687401000535379625274114368932750282085652829625752508001128621612480632332
132062530751124411925269858862037532094603351079875201064792582488120023096673643306263523022538038014820247114248970752
148188092004863230400128419677751988433720020050040010138276138006666885225225756960053183130085093008886385556056019110280181836821324
Technical details
Node id
n-2741ba01bd46e749 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7ffe7dd0b586de9b

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:      7,       1;
  2:     36,      16,       1;
  3:    165,     135,      27,       1;
  4:    715,     880,     330,      40,       1;
  5:   3003,    5005,    2860,     660,      55,      1;
  6:  12376,   26208,   20475,    7280,    1170,     72,     1;
  7:  50388,  129948,  129948,   63700,   15925,   1911,    91,    1;
  8: 203490,  620160,  759696,  479808,  166600,  31360,  2940,  112,   1;
  9: 817190, 2877930, 4186080, 3255840, 1465128, 385560, 57120, 4320, 135, 1;
%F Formula
The row polynomials satisfy (n^2+5*n)*P(n,x) = (4*n^2+18*n+20+(3*n^2+8*n-5)*x)*P(n-1,x) - ((2*n+5)*x+(2*n+5)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 18*n + 20 + (3*n^2 + 8*n - 5)*x)*P[n - 1, x] - ((2*n + 5)*x + (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 18*n + 20 + (3*n^2 + 8*n - 5)*x)*v[i - 1] - ((2*n + 5)*x + (2*n + 5)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n)); 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. [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.