Real-rooted atlas

A008459 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 11\,n x + 4\,n x^{2} - 6\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(4 + 30\,x + 48\,x^{2} - 32\,x^{3} - 4\,n - 56\,n x - 132\,n x^{2} + 48\,n x^{3} - 16\,n^{2} - 40\,n^{2} x + 96\,n^{2} x^{2} + 16\,n^{3} + 48\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 14\,x^{2} + 16\,x^{3} - 32\,x^{4} - 16\,n x^{2} - 64\,n x^{3} - 8\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2305616102
3140504384641092
463034405280230425611910
52772204404928044160128001024130476
6120121118883729605376003225606758440961428700
75148058027224837125040000494592021504003440641638415612232

All 15 rows

Technical details
Node id
n-dd14aeade0002033 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-16e44bdaa85768a5

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      6,        4;
  2:     30,       56,       16;
  3:    140,      504,      384,        64;
  4:    630,     3440,     5280,      2304,       256;
  5:   2772,    20440,    49280,     44160,     12800,      1024;
  6:  12012,   111888,   372960,    537600,    322560,     67584,      4096;
  7:  51480,   580272,  2483712,   5040000,   4945920,   2150400,    344064,    16384;
  8: 218790,  2896608, 15183168,  40094208,  56179200,  40427520,  13418496,  1703936,   65536;
  9: 923780, 14054040, 87310080, 285212928, 525837312, 546416640, 302776320, 79626240, 8257536, 262144;
%F Formula
The row polynomials satisfy (4*n^3-6*n^2+2*n+(8*n^2-11*n)*x+4*n*x^2)*P(n,x) = (16*n^3-16*n^2-4*n+4+(48*n^3-40*n^2-56*n+30)*x+(96*n^2-132*n+48)*x^2+(48*n-32)*x^3)*P(n-1,x) - ((8*n^2-2)*x+(32*n^2+16*n-14)*x^2+(64*n-16)*x^3+32*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 16*n^2 - 4*n + 4 + (48*n^3 - 40*n^2 - 56*n + 30)*x + (96*n^2 - 132*n + 48)*x^2 + (48*n - 32)*x^3)*P[n - 1, x] - ((8*n^2 - 2)*x + (32*n^2 + 16*n - 14)*x^2 + (64*n - 16)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 + 2*n + (8*n^2 - 11*n)*x + 4*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 - 16*n^2 - 4*n + 4 + (48*n^3 - 40*n^2 - 56*n + 30)*x + (96*n^2 - 132*n + 48)*x^2 + (48*n - 32)*x^3)*v[i - 1] - ((8*n^2 - 2)*x + (32*n^2 + 16*n - 14)*x^2 + (64*n - 16)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 + 2*n + (8*n^2 - 11*n)*x + 4*n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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