Real-rooted atlas

narayana → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 22\,x - 58\,x^{2} - 58\,x^{3} - 20\,x^{4} + 4\,n + 26\,n x + 50\,n x^{2} + 38\,n x^{3} + 10\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 16\,x^{2} - 36\,x^{3} - 32\,x^{4} - 10\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2268
35304176
414126354306800
5425042156386424268992
6132198011400313204082020076105728
74297722559352079004164754246021714811284544
81430300302622621231230334565052460104382210150066615999488
94862116688119319267747682330728449633584637442964504526413386206203206144
10167964534925314608354201121476755283986826486956607847551891364620830041212674762621763584
115878617635802330445017837352087445722028646850966341293140935199720087818448904735764540111267402634266216448
12208012686439610097626087297210049223262001898333791251030611016955000846801217872887001006853463004852194085210318939956452730896384
1374290026744400433437576417747528026640038700118294434720374651911920851815093536137845068174015468863652001141663877640497195969040965721689166036808941568
1426744401043031601846552240196321725601397283762167023983400722563321002528686978826019213508833643448192336571936081925748540830412831102934416509629543082491089699285681135093284864
159694845407183490781839103590900203940714380551885401143289547016584300079347512412706943281188079944321272055569769833582610545477229692358283792049641431706849391915226066681179486505666014011097980336390144
Technical details
Node id
n-cfd156b289caa31b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0ce25fdf433dc6d0

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     2,      6;
   3:     5,     30,      41;
   4:    14,    126,     354,      306;
   5:    42,    504,    2156,     3864,      2426;
   6:   132,   1980,   11400,    31320,     40820,     20076;
   7:   429,   7722,   55935,   207900,    416475,    424602,    171481;
   8:  1430,  30030,  262262,  1231230,   3345650,   5246010,   4382210,   1500666;
   9:  4862, 116688, 1193192,  6774768,  23307284,  49633584,  63744296,  45045264,  13386206;
  10: 16796, 453492, 5314608, 35420112, 147675528, 398682648, 695660784, 755189136, 462083004, 121267476;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (4*n-2+(26*n-22)*x+(50*n-58)*x^2+(38*n-58)*x^3+(10*n-20)*x^4)*P(n-1,x) - (2*x+16*x^2+36*x^3+32*x^4+10*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 2 + (26*n - 22)*x + (50*n - 58)*x^2 + (38*n - 58)*x^3 + (10*n - 20)*x^4)*P[n - 1, x] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*D[P[n - 1, x], x])/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 2 + (26*n - 22)*x + (50*n - 58)*x^2 + (38*n - 58)*x^3 + (10*n - 20)*x^4)*v[i - 1] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*deriv(v[i - 1], x))/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023