Real-rooted atlas

narayana → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 7\,x + 2\,x^{2} - 2\,n - 3\,n x + 2\,n x^{2} - 2\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(12\,x + 26\,x^{2} - 12\,x^{3} + 4\,n - 18\,n x - 46\,n x^{2} + 12\,n x^{3} - 12\,n^{2} - 12\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x^{2} + 8\,x^{3} - 8\,x^{4} + 2\,n x - 4\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31518437
456114608238
5210616560176161578
67923060408022404803210684
73003144542574021000792012486473429
8114406606614814816324092400257603136128510318
94375829515280080011211208648643646727884876802563577150
101679601296828413712070586886971328403603213278722304001843251225245172
11646646562666020659080416894405049408037669632171225604542720648960435201024179144322
122496144241746121004788402344980003376771203106414081829667846741504014784000177408010137620481277009452
139657700103055088478546464126941584021232728002333041920170271129681688780824989184046182400473088023347240969137631604
1437442160436528040224058640066630403841271416889616291045056142602071048513556480340542259288122777613940326412353536532480819265595522360
151454226751839346110103433967003409940716073174461360107322543328109940166336790164460803948161817613408851456297974476840870502431682560120422416384472193012341
Technical details
Node id
n-4c13043f90815ef6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4372db64edf115ba

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:      4,       2;
   3:     15,      18,       4;
   4:     56,     114,      60,       8;
   5:    210,     616,     560,     176,      16;
   6:    792,    3060,    4080,    2240,     480,      32;
   7:   3003,   14454,   25740,   21000,    7920,    1248,      64;
   8:  11440,   66066,  148148,  163240,   92400,   25760,    3136,    128;
   9:  43758,  295152,  800800, 1121120,  864864,  364672,   78848,   7680,   256;
  10: 167960, 1296828, 4137120, 7058688, 6971328, 4036032, 1327872, 230400, 18432, 512;
%F Formula
The row polynomials satisfy (2*n^3-2*n^2-2*n+2+(4*n^2-3*n-7)*x+(2*n+2)*x^2)*P(n,x) = (8*n^3-12*n^2+4*n+(12*n^3-12*n^2-18*n+12)*x+(24*n^2-46*n+26)*x^2+(12*n-12)*x^3)*P(n-1,x) - ((4*n^2-2*n)*x+(8*n^2+4*n-6)*x^2+(16*n-8)*x^3+8*x^4)*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[((8*n^3 - 12*n^2 + 4*n + (12*n^3 - 12*n^2 - 18*n + 12)*x + (24*n^2 - 46*n + 26)*x^2 + (12*n - 12)*x^3)*P[n - 1, x] - ((4*n^2 - 2*n)*x + (8*n^2 + 4*n - 6)*x^2 + (16*n - 8)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 2*n^2 - 2*n + 2 + (4*n^2 - 3*n - 7)*x + (2*n + 2)*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] = ((8*n^3 - 12*n^2 + 4*n + (12*n^3 - 12*n^2 - 18*n + 12)*x + (24*n^2 - 46*n + 26)*x^2 + (12*n - 12)*x^3)*v[i - 1] - ((4*n^2 - 2*n)*x + (8*n^2 + 4*n - 6)*x^2 + (16*n - 8)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 2*n^2 - 2*n + 2 + (4*n^2 - 3*n - 7)*x + (2*n + 2)*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