Real-rooted atlas

A062264 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(6 + 6\,x + 10\,n + 6\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
155
2423072
3252336105693
41320252015122805632
564351584013860504063041805
6300309009010296055440138601260293640
71361364804806756754804801801803326423101988525
8604656245044840840803603600180180050450472072396013125120
926453701209312023279256245044801531530057657601261260144144643585015125
1011440660581981401269777601551950401163962805513508016336320288288027027010010542842440
1149031400274575840669278610931170240814773960465585120174594420420076806126120480480150153427638885
1220860632012748164003432198000535422888053542288803585005424162954792049884120099768240122522408168162184021450332160
13882230895584097696017210021400297457160003346393050025700298624137425207925121436320130945815022170720023279256133660830940133262943645
14371255535026466926850846941659201606268664002007835830001740124386001070845776004711721414414724129420320089770046558512042325920211629642840822933425160
Technical details
Node id
n-ae7fa795b878b77c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-51638b1898cb1bac

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:       5;
  2:      42,       30;
  3:     252,      336,      105;
  4:    1320,     2520,     1512,      280;
  5:    6435,    15840,    13860,     5040,      630;
  6:   30030,    90090,   102960,    55440,    13860,    1260;
  7:  136136,   480480,   675675,   480480,   180180,   33264,    2310;
  8:  604656,  2450448,  4084080,  3603600,  1801800,  504504,   72072,   3960;
  9: 2645370, 12093120, 23279256, 24504480, 15315300, 5765760, 1261260, 144144, 6435;
%F Formula
The row polynomials satisfy (n^2+2*n-3)*P(n,x) = (4*n^2+10*n+6+(3*n^2+6*n+6)*x)*P(n-1,x) - ((2*n+3)*x+(2*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 6)*x)*P[n - 1, x] - ((2*n + 3)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 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] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 6)*x)*v[i - 1] - ((2*n + 3)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 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. [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.