Real-rooted atlas

narayana → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n x + n^{2} - 8\,n^{2} x + 4\,n^{2} x^{2} + n^{3} - 3\,n^{3} x - 4\,n^{3} x^{2} + 6\,n^{4} x - 4\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(16\,x^{2} - 32\,x^{3} - 2\,n x + 56\,n x^{2} + 48\,n x^{3} - 2\,n^{2} - 26\,n^{2} x - 48\,n^{2} x^{2} + 48\,n^{2} x^{3} + 4\,n^{3} - 120\,n^{3} x^{2} - 96\,n^{3} x^{3} + 24\,n^{4} x + 80\,n^{4} x^{2} - 16\,n^{4} x^{3} + 16\,n^{5} x^{2} + 48\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} - 16\,x^{4} + 5\,n x^{2} + 24\,n x^{3} + 16\,n x^{4} - 2\,n^{2} x - 8\,n^{2} x^{2} + 8\,n^{2} x^{3} + 32\,n^{2} x^{4} - 12\,n^{3} x^{2} - 56\,n^{3} x^{3} - 32\,n^{3} x^{4} - 8\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221618
3580144229
414336129610242670
5421344806414336640030186
613252804320012288012800036864336356
74292059221384084480014400009953282007043715693
81430800801009008512512012320000141926407024640104857640801494

All 15 rows

Technical details
Node id
n-035bf07f85d18538 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-777088b2ee260e77

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,      16;
   3:     5,      80,      144;
   4:    14,     336,     1296,      1024;
   5:    42,    1344,     8064,     14336,      6400;
   6:   132,    5280,    43200,    122880,    128000,      36864;
   7:   429,   20592,   213840,    844800,   1440000,     995328,     200704;
   8:  1430,   80080,  1009008,   5125120,  12320000,   14192640,    7024640,   1048576;
   9:  4862,  311168,  4612608,  28700672,  89689600,  147603456,  123633664,  46137344,   5308416;
  10: 16796, 1209312, 20621952, 152076288, 587059200, 1268047872, 1534181376, 981467136, 286654464, 26214400;
%F Formula
The row polynomials satisfy (n^3+n^2+(6*n^4-3*n^3-8*n^2+n)*x+(4*n^5-4*n^4-4*n^3+4*n^2)*x^2)*P(n,x) = (4*n^3-2*n^2+(24*n^4-26*n^2-2*n)*x+(16*n^5+80*n^4-120*n^3-48*n^2+56*n+16)*x^2+(48*n^5-16*n^4-96*n^3+48*n^2+48*n-32)*x^3)*P(n-1,x) - (2*n^2*x+(12*n^3+8*n^2-5*n-1)*x^2+(8*n^4+56*n^3-8*n^2-24*n)*x^3+(32*n^4+32*n^3-32*n^2-16*n+16)*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[((4*n^3 - 2*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (16*n^5 + 80*n^4 - 120*n^3 - 48*n^2 + 56*n + 16)*x^2 + (48*n^5 - 16*n^4 - 96*n^3 + 48*n^2 + 48*n - 32)*x^3)*P[n - 1, x] - (2*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (8*n^4 + 56*n^3 - 8*n^2 - 24*n)*x^3 + (32*n^4 + 32*n^3 - 32*n^2 - 16*n + 16)*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*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] = ((4*n^3 - 2*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (16*n^5 + 80*n^4 - 120*n^3 - 48*n^2 + 56*n + 16)*x^2 + (48*n^5 - 16*n^4 - 96*n^3 + 48*n^2 + 48*n - 32)*x^3)*v[i - 1] - (2*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (8*n^4 + 56*n^3 - 8*n^2 - 24*n)*x^3 + (32*n^4 + 32*n^3 - 32*n^2 - 16*n + 16)*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*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