Real-rooted atlas

A094638 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
266113
3243312170
4120200105201446
572013709002553013276
65040105848120294052542127252
7403209147678792338457840966561253296
83628808766728268684037041122451814416387212602224
93628800923918493816005065760161595031636537800261090129288160
1039916800106286400114782184672760002391851054123307888657260039601101358457760
114790016001325982240150917976094732268436796584093376745158253481787115130680577513214740577920
126227020800178373145602124715507214140148880591486152416485612003149316174155980437473152230808151156167375532160
13871782912002574358675203191606046722234512840569957703756030025069074632754522494525531110001791873723653643641119318211024208720640
14130767436800039685190630405098038371712373187112480017569376624805666336676001296839063522145163020025788813052238736501374873557330015015210116583626886400
Technical details
Node id
n-0d0c19e5a4636085 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-7d8c365492dbd6da

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,       1;
  2:       6,       6,       1;
  3:      24,      33,      12,       1;
  4:     120,     200,     105,      20,       1;
  5:     720,    1370,     900,     255,      30,      1;
  6:    5040,   10584,    8120,    2940,     525,     42,     1;
  7:   40320,   91476,   78792,   33845,    7840,    966,    56,    1;
  8:  362880,  876672,  826868,  403704,  112245,  18144,  1638,   72,  1;
  9: 3628800, 9239184, 9381600, 5065760, 1615950, 316365, 37800, 2610, 90, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3-2*n^2+n+(2*n^2-2*n)*x+n*x^2)*P(n-2,x) + (2*n^2-2*n+(2*n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 2*n^2 + n + (2*n^2 - 2*n)*x + n*x^2)*P[n - 2, x] + (2*n^2 - 2*n + (2*n - 1)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 2*n^2 + n + (2*n^2 - 2*n)*x + n*x^2)*v[i - 2] + (2*n^2 - 2*n + (2*n - 1)*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

Same fingerprint ?

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.