Real-rooted atlas

A207606 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-810\,x - 1782\,x^{2} + 972\,x^{3} - 60\,n - 594\,n x - 1296\,n x^{2} + 2106\,n x^{3} + 1458\,n^{2} x^{3} + 54\,n^{3} x^{2} + 324\,n^{3} x^{3}\right)\,P_{n}(x) = \left(60 - 1206\,x - 7182\,x^{2} - 5184\,x^{3} + 2916\,x^{4} - 90\,n - 1071\,n x - 4941\,n x^{2} - 162\,n x^{3} + 5346\,n x^{4} - 45\,n^{2} x - 324\,n^{2} x^{2} + 2349\,n^{2} x^{3} + 2916\,n^{2} x^{4} + 81\,n^{3} x^{2} + 567\,n^{3} x^{3} + 486\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 435\,x - 2187\,x^{2} - 3051\,x^{3} + 1296\,x^{4} + 1458\,x^{5} + 45\,n x^{2} + 1242\,n x^{3} + 3969\,n x^{4} + 1944\,n x^{5} + 45\,n^{2} x^{2} + 432\,n^{2} x^{3} + 1053\,n^{2} x^{4} + 486\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 66\,x + 54\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0136654133
134264459216973
289930251124848106824
32333060112591728011745291646493
461096484503694392992255103010206310147
5159729544167265446040639900507384209223349922035945
6418188566589707191278834931253774762239330782231211809813196846
7109462612402000403765115217012430233921522018236510637568312959739366084671513

All 15 rows

Technical details
Node id
n-7a55accc66fefeec · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-847bbfa887a5319d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:    13,      66,       54;
  1:    34,     264,      459,       216;
  2:    89,     930,     2511,      2484,       810;
  3:   233,    3060,    11259,     17280,     11745,      2916;
  4:   610,    9648,    45036,     94392,     99225,     51030,     10206;
  5:  1597,   29544,   167265,    446040,    639900,    507384,    209223,     34992;
  6:  4181,   88566,   589707,   1912788,   3493125,   3774762,   2393307,    822312,    118098;
  7: 10946,  261240,  2000403,   7651152,  17012430,  23392152,  20182365,  10637568,   3129597,    393660;
  8: 28657,  760830,  6587082,  29042280,  76235985, 127646442, 140026320, 100287072,  45172485,  11612970,  1299078;
  9: 75025, 2193192, 21184578, 105824016, 320604075, 633996720, 846659142, 770313888, 470856726, 185020200, 42220035, 4251528;
%F Formula
The row polynomials satisfy -(60*n+(594*n+810)*x-(54*n^3-1296*n-1782)*x^2-(324*n^3+1458*n^2+2106*n+972)*x^3)*P(n,x) = -(90*n-60+(45*n^2+1071*n+1206)*x-(81*n^3-324*n^2-4941*n-7182)*x^2-(567*n^3+2349*n^2-162*n-5184)*x^3-(486*n^3+2916*n^2+5346*n+2916)*x^4)*P(n-1,x) - (25+435*x-(45*n^2+45*n-2187)*x^2-(432*n^2+1242*n-3051)*x^3-(1053*n^2+3969*n+1296)*x^4-(486*n^2+1944*n+1458)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 66*x + 54*x^2.
%t Mathematica
P[0, x_] := 13 + 66*x + 54*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(90*n - 60 + (45*n^2 + 1071*n + 1206)*x - (81*n^3 - 324*n^2 - 4941*n - 7182)*x^2 - (567*n^3 + 2349*n^2 - 162*n - 5184)*x^3 - (486*n^3 + 2916*n^2 + 5346*n + 2916)*x^4)*P[n - 1, x] - (25 + 435*x - (45*n^2 + 45*n - 2187)*x^2 - (432*n^2 + 1242*n - 3051)*x^3 - (1053*n^2 + 3969*n + 1296)*x^4 - (486*n^2 + 1944*n + 1458)*x^5)*D[P[n - 1, x], x])/(-60*n - (594*n + 810)*x + (54*n^3 - 1296*n - 1782)*x^2 + (324*n^3 + 1458*n^2 + 2106*n + 972)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 66*x + 54*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(90*n - 60 + (45*n^2 + 1071*n + 1206)*x - (81*n^3 - 324*n^2 - 4941*n - 7182)*x^2 - (567*n^3 + 2349*n^2 - 162*n - 5184)*x^3 - (486*n^3 + 2916*n^2 + 5346*n + 2916)*x^4)*v[i - 1] - (25 + 435*x - (45*n^2 + 45*n - 2187)*x^2 - (432*n^2 + 1242*n - 3051)*x^3 - (1053*n^2 + 3969*n + 1296)*x^4 - (486*n^2 + 1944*n + 1458)*x^5)*deriv(v[i - 1], x))/(-60*n - (594*n + 810)*x + (54*n^3 - 1296*n - 1782)*x^2 + (324*n^3 + 1458*n^2 + 2106*n + 972)*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