Real-rooted atlas

A207607 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(150 + 900\,x + 1458\,x^{2} - 60\,n - 684\,n x - 1107\,n x^{2} + 648\,n x^{3} - 405\,n^{2} x^{2} - 972\,n^{2} x^{3} + 54\,n^{3} x^{2} + 324\,n^{3} x^{3}\right)\,P_{n}(x) = \left(360 + 2754\,x + 6480\,x^{2} + 4536\,x^{3} + 972\,x^{4} - 90\,n - 981\,n x - 2970\,n x^{2} - 2754\,n x^{3} - 486\,n x^{4} - 45\,n^{2} x - 999\,n^{2} x^{2} - 2349\,n^{2} x^{3} - 972\,n^{2} x^{4} + 81\,n^{3} x^{2} + 567\,n^{3} x^{3} + 486\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 510\,x - 2997\,x^{2} - 6480\,x^{3} - 4536\,x^{4} - 972\,x^{5} - 180\,n x^{2} - 918\,n x^{3} - 1296\,n x^{4} - 486\,n x^{5} + 45\,n^{2} x^{2} + 432\,n^{2} x^{3} + 1053\,n^{2} x^{4} + 486\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
285454116
321198405216840
455666205222688105851
5144213087481479610935291639669
637765883377777112874804811410206263654
798719896122229351648540675456354199017349921725798

All 15 rows

Technical details
Node id
n-d06e1fa708981619 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e75b70736cdfa66a

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,     54,      54;
  3:   21,    198,     405,     216;
  4:   55,    666,    2052,    2268,      810;
  5:  144,   2130,    8748,   14796,    10935,     2916;
  6:  377,   6588,   33777,   77112,    87480,    48114,    10206;
  7:  987,  19896,  122229,  351648,   540675,   456354,   199017,   34992;
  8: 2584,  59022,  422442, 1466748,  2853225,  3267378,  2184084,  787320,  118098;
  9: 6765, 172674, 1410696, 5738364, 13519305, 19617390, 17789058, 9815256, 3011499, 393660;
%F Formula
The row polynomials satisfy -(60*n-150+(684*n-900)*x-(54*n^3-405*n^2-1107*n+1458)*x^2-(324*n^3-972*n^2+648*n)*x^3)*P(n,x) = -(90*n-360+(45*n^2+981*n-2754)*x-(81*n^3-999*n^2-2970*n+6480)*x^2-(567*n^3-2349*n^2-2754*n+4536)*x^3-(486*n^3-972*n^2-486*n+972)*x^4)*P(n-1,x) - (25+510*x-(45*n^2-180*n-2997)*x^2-(432*n^2-918*n-6480)*x^3-(1053*n^2-1296*n-4536)*x^4-(486*n^2-486*n-972)*x^5)*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[(-(90*n - 360 + (45*n^2 + 981*n - 2754)*x - (81*n^3 - 999*n^2 - 2970*n + 6480)*x^2 - (567*n^3 - 2349*n^2 - 2754*n + 4536)*x^3 - (486*n^3 - 972*n^2 - 486*n + 972)*x^4)*P[n - 1, x] - (25 + 510*x - (45*n^2 - 180*n - 2997)*x^2 - (432*n^2 - 918*n - 6480)*x^3 - (1053*n^2 - 1296*n - 4536)*x^4 - (486*n^2 - 486*n - 972)*x^5)*D[P[n - 1, x], x])/(-60*n + 150 - (684*n - 900)*x + (54*n^3 - 405*n^2 - 1107*n + 1458)*x^2 + (324*n^3 - 972*n^2 + 648*n)*x^3)]];
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] = (-(90*n - 360 + (45*n^2 + 981*n - 2754)*x - (81*n^3 - 999*n^2 - 2970*n + 6480)*x^2 - (567*n^3 - 2349*n^2 - 2754*n + 4536)*x^3 - (486*n^3 - 972*n^2 - 486*n + 972)*x^4)*v[i - 1] - (25 + 510*x - (45*n^2 - 180*n - 2997)*x^2 - (432*n^2 - 918*n - 6480)*x^3 - (1053*n^2 - 1296*n - 4536)*x^4 - (486*n^2 - 486*n - 972)*x^5)*deriv(v[i - 1], x))/(-60*n + 150 - (684*n - 900)*x + (54*n^3 - 405*n^2 - 1107*n + 1458)*x^2 + (324*n^3 - 972*n^2 + 648*n)*x^3)); 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