Real-rooted atlas

A063007 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 27\,x - 54\,x^{2} - 12\,n - 27\,n x + 135\,n x^{2} - 108\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(36 + 153\,x + 162\,x^{2} - 36\,n - 117\,n x + 81\,n x^{2} + 324\,n x^{3} - 36\,n^{2} x - 324\,n^{2} x^{2} - 486\,n^{2} x^{3} + 81\,n^{3} x^{2} + 162\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 72\,x - 198\,x^{2} - 162\,x^{3} - 72\,n x^{2} - 324\,n x^{3} - 324\,n x^{4} + 36\,n^{2} x^{2} + 162\,n^{2} x^{3} + 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
231215
313108162283
463792243021605445
53215400251104536028350104541
61683354602211306048007654503674161995939
78989227556178038065318401250235012124728471517237891015
8486391437744135331566232464015997905023441140818389170860046272715672617

All 15 rows

Technical details
Node id
n-26a108817d80da69 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-70cbdb650a08f4fc

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:       3,       12;
   3:      13,      108,       162;
   4:      63,      792,      2430,       2160;
   5:     321,     5400,     25110,      45360,       28350;
   6:    1683,    35460,    221130,     604800,      765450,      367416;
   7:    8989,   227556,   1780380,    6531840,    12502350,    12124728,     4715172;
   8:   48639,  1437744,  13533156,   62324640,   159979050,   234411408,   183891708,    60046272;
   9:  265729,  8982576,  98828100,  547858080,  1765382850,  3467672208,  4106914812,  2702082240,   759960630;
  10: 1462563, 55646460, 700700220, 4542531840, 17622955350, 43450983576, 68959390500, 68452750080, 38757992130, 9569874600;
%F Formula
The row polynomials satisfy -(12*n-12+(27*n-27)*x-(27*n^3-108*n^2+135*n-54)*x^2)*P(n,x) = -(36*n-36+(36*n^2+117*n-153)*x-(81*n^3-324*n^2+81*n+162)*x^2-(162*n^3-486*n^2+324*n)*x^3)*P(n-1,x) - (8+72*x-(36*n^2-72*n-198)*x^2-(162*n^2-324*n-162)*x^3-(162*n^2-324*n)*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[(-(36*n - 36 + (36*n^2 + 117*n - 153)*x - (81*n^3 - 324*n^2 + 81*n + 162)*x^2 - (162*n^3 - 486*n^2 + 324*n)*x^3)*P[n - 1, x] - (8 + 72*x - (36*n^2 - 72*n - 198)*x^2 - (162*n^2 - 324*n - 162)*x^3 - (162*n^2 - 324*n)*x^4)*D[P[n - 1, x], x])/(-12*n + 12 - (27*n - 27)*x + (27*n^3 - 108*n^2 + 135*n - 54)*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] = (-(36*n - 36 + (36*n^2 + 117*n - 153)*x - (81*n^3 - 324*n^2 + 81*n + 162)*x^2 - (162*n^3 - 486*n^2 + 324*n)*x^3)*v[i - 1] - (8 + 72*x - (36*n^2 - 72*n - 198)*x^2 - (162*n^2 - 324*n - 162)*x^3 - (162*n^2 - 324*n)*x^4)*deriv(v[i - 1], x))/(-12*n + 12 - (27*n - 27)*x + (27*n^3 - 108*n^2 + 135*n - 54)*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

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