Real-rooted atlas

narayana → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(3\,n x + 4\,n^{2} - 24\,n^{2} x + 9\,n^{2} x^{2} + 4\,n^{3} - 9\,n^{3} x - 9\,n^{3} x^{2} + 18\,n^{4} x - 9\,n^{4} x^{2} + 9\,n^{5} x^{2}\right)\,P_{n}(x) = \left(36\,x^{2} - 54\,x^{3} - 6\,n x + 126\,n x^{2} + 81\,n x^{3} - 8\,n^{2} - 78\,n^{2} x - 108\,n^{2} x^{2} + 81\,n^{2} x^{3} + 16\,n^{3} - 270\,n^{3} x^{2} - 162\,n^{3} x^{3} + 72\,n^{4} x + 180\,n^{4} x^{2} - 27\,n^{4} x^{3} + 36\,n^{5} x^{2} + 81\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(3\,x^{2} - 27\,x^{4} + 15\,n x^{2} + 54\,n x^{3} + 27\,n x^{4} - 8\,n^{2} x - 24\,n^{2} x^{2} + 18\,n^{2} x^{3} + 54\,n^{2} x^{4} - 36\,n^{3} x^{2} - 126\,n^{3} x^{3} - 54\,n^{3} x^{4} - 18\,n^{4} x^{3} - 54\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221214
356081146
4142527294321427
542100845366048202513659
613239602430051840405008748129480
742915444120285356400455625236196357211220100
8143060060567567216216038981253367980125023513996811447525

All 15 rows

Technical details
Node id
n-7ddb81b454012205 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-db52feb71393f616

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,     12;
   3:     5,     60,       81;
   4:    14,    252,      729,      432;
   5:    42,   1008,     4536,     6048,      2025;
   6:   132,   3960,    24300,    51840,     40500,      8748;
   7:   429,  15444,   120285,   356400,    455625,    236196,     35721;
   8:  1430,  60060,   567567,  2162160,   3898125,   3367980,   1250235,    139968;
   9:  4862, 233376,  2594592, 12108096,  28378350,  35026992,  22004136,   6158592,   531441;
  10: 16796, 906984, 11599848, 64157184, 185749200, 300913704, 273051324, 131010048, 28697814, 1968300;
%F Formula
The row polynomials satisfy (4*n^3+4*n^2+(18*n^4-9*n^3-24*n^2+3*n)*x+(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n,x) = (16*n^3-8*n^2+(72*n^4-78*n^2-6*n)*x+(36*n^5+180*n^4-270*n^3-108*n^2+126*n+36)*x^2+(81*n^5-27*n^4-162*n^3+81*n^2+81*n-54)*x^3)*P(n-1,x) - (8*n^2*x+(36*n^3+24*n^2-15*n-3)*x^2+(18*n^4+126*n^3-18*n^2-54*n)*x^3+(54*n^4+54*n^3-54*n^2-27*n+27)*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[((16*n^3 - 8*n^2 + (72*n^4 - 78*n^2 - 6*n)*x + (36*n^5 + 180*n^4 - 270*n^3 - 108*n^2 + 126*n + 36)*x^2 + (81*n^5 - 27*n^4 - 162*n^3 + 81*n^2 + 81*n - 54)*x^3)*P[n - 1, x] - (8*n^2*x + (36*n^3 + 24*n^2 - 15*n - 3)*x^2 + (18*n^4 + 126*n^3 - 18*n^2 - 54*n)*x^3 + (54*n^4 + 54*n^3 - 54*n^2 - 27*n + 27)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 4*n^2 + (18*n^4 - 9*n^3 - 24*n^2 + 3*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*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] = ((16*n^3 - 8*n^2 + (72*n^4 - 78*n^2 - 6*n)*x + (36*n^5 + 180*n^4 - 270*n^3 - 108*n^2 + 126*n + 36)*x^2 + (81*n^5 - 27*n^4 - 162*n^3 + 81*n^2 + 81*n - 54)*x^3)*v[i - 1] - (8*n^2*x + (36*n^3 + 24*n^2 - 15*n - 3)*x^2 + (18*n^4 + 126*n^3 - 18*n^2 - 54*n)*x^3 + (54*n^4 + 54*n^3 - 54*n^2 - 27*n + 27)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 4*n^2 + (18*n^4 - 9*n^3 - 24*n^2 + 3*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*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