Real-rooted atlas

narayana → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-14 - 12\,x + 16\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-10 - 18\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
26410
3415216109
4306564336641270
5242658405120192025615562
620076595606880038400102401024198100
71714816042848656806416002572805222440962596645
815006666122452104667369674560518784015984642580481638434825150
9133862066204438412324121613650918491753984380477449404416124518465536475697854
101212674766293242801424154240183880704014867435527763374082597683205308416058982402621446595646860
1111126740266390975720162302952002393612160022626293760142619688966034882560167854080028999680027525120104857692590323058
1210318939956649863986641830076353603035185824003284915251202427376680961243880816644392419328010380902400154271744012687769641943041313427716380
13965721689166616700579522046271366656376985127552045972366105603898034528256234767135539210069734850563032521113606193938432080278978565788139521677721618798095833012
1491089699285667453654763042272552320729646045688392704624758196142085979741345177641409755971584209249406812167675829551104200463299379235867171225641003515904261724569667108864271118225915560
1586505666014016884726560782025098357711336055469407165536082887404228428888403589616742469222964242432040297490128896017453715062784055669725855744127770799636482024821227520206108098560117440512002684354563936516861402901

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 + n\right)\,P_{n}(x) = \left(32 + 64\,x + 32\,x^{2} - 16\,n - 32\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 4\,x + 12\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)
Technical details
Node id
n-1a9e5fc6991d4de9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-10570a71b7b0777d)
Fingerprint
f6-ea13047c46098f9c

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:         6,         4;
   3:        41,        52,         16;
   4:       306,       564,        336,         64;
   5:      2426,      5840,       5120,       1920,        256;
   6:     20076,     59560,      68800,      38400,      10240,      1024;
   7:    171481,    604284,     865680,     641600,     257280,     52224,      4096;
   8:   1500666,   6122452,   10466736,    9674560,    5187840,   1598464,    258048,    16384;
   9:  13386206,  62044384,  123241216,  136509184,   91753984,  38047744,   9404416,  1245184,   65536;
  10: 121267476, 629324280, 1424154240, 1838807040, 1486743552, 776337408, 259768320, 53084160, 5898240, 262144;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (16*n-14+(12*n-12)*x)*P(n-1,x) - (10+18*x+8*x^2)*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[((16*n - 14 + (12*n - 12)*x)*P[n - 1, x] - (10 + 18*x + 8*x^2)*D[P[n - 1, x], x])/(n + 1)]];
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] = ((16*n - 14 + (12*n - 12)*x)*v[i - 1] - (10 + 18*x + 8*x^2)*deriv(v[i - 1], x))/(n + 1)); 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