Real-rooted atlas

A172431 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(60\,n + 126\,n x + 180\,n^{2} x + 108\,n^{2} x^{2} + 216\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-60 - 396\,x - 756\,x^{2} - 324\,x^{3} + 240\,n + 1134\,n x + 1080\,n x^{2} - 648\,n x^{3} + 900\,n^{2} x + 3456\,n^{2} x^{2} + 2916\,n^{2} x^{3} + 864\,n^{3} x^{2} + 1944\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 240\,x - 837\,x^{2} - 1242\,x^{3} - 648\,x^{4} - 150\,n x - 1170\,n x^{2} - 3024\,n x^{3} - 2592\,n x^{4} - 180\,n^{2} x^{2} - 972\,n^{2} x^{3} - 1296\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
286081149
3212285944321275
45578029974320202510177
51442520127982764826325874878183
637778484957214299220695514288435721586349
798723820179982650160126765013471927144201399684324179

All 15 rows

Technical details
Node id
n-36f89b00238f1d0b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5e3fef85e54d325c

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:    3,     12;
  2:    8,     60,      81;
  3:   21,    228,     594,      432;
  4:   55,    780,    2997,     4320,     2025;
  5:  144,   2520,   12798,    27648,    26325,     8748;
  6:  377,   7848,   49572,   142992,   206955,   142884,    35721;
  7:  987,  23820,  179982,   650160,  1267650,  1347192,   714420,   139968;
  8: 2584,  70932,  623943,  2709504,  6656175,  9605304,  7930062,  3359232,   531441;
  9: 6765, 208140, 2089206, 10601280, 31446225, 57488940, 64910160, 43250112, 15057495, 1968300;
%F Formula
The row polynomials satisfy (60*n+(180*n^2+126*n)*x+(216*n^3+108*n^2)*x^2)*P(n,x) = (240*n-60+(900*n^2+1134*n-396)*x+(864*n^3+3456*n^2+1080*n-756)*x^2+(1944*n^3+2916*n^2-648*n-324)*x^3)*P(n-1,x) - (25+(150*n+240)*x+(180*n^2+1170*n+837)*x^2+(972*n^2+3024*n+1242)*x^3+(1296*n^2+2592*n+648)*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[((240*n - 60 + (900*n^2 + 1134*n - 396)*x + (864*n^3 + 3456*n^2 + 1080*n - 756)*x^2 + (1944*n^3 + 2916*n^2 - 648*n - 324)*x^3)*P[n - 1, x] - (25 + (150*n + 240)*x + (180*n^2 + 1170*n + 837)*x^2 + (972*n^2 + 3024*n + 1242)*x^3 + (1296*n^2 + 2592*n + 648)*x^4)*D[P[n - 1, x], x])/(60*n + (180*n^2 + 126*n)*x + (216*n^3 + 108*n^2)*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] = ((240*n - 60 + (900*n^2 + 1134*n - 396)*x + (864*n^3 + 3456*n^2 + 1080*n - 756)*x^2 + (1944*n^3 + 2916*n^2 - 648*n - 324)*x^3)*v[i - 1] - (25 + (150*n + 240)*x + (180*n^2 + 1170*n + 837)*x^2 + (972*n^2 + 3024*n + 1242)*x^3 + (1296*n^2 + 2592*n + 648)*x^4)*deriv(v[i - 1], x))/(60*n + (180*n^2 + 126*n)*x + (216*n^3 + 108*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 - 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