Real-rooted atlas

A207607 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(21 + 72\,x + 459\,x^{2} + 324\,x^{3} + 267\,n + 576\,n x - 378\,n x^{2} - 324\,n x^{3} - 405\,n^{2} - 972\,n^{2} x + 54\,n^{3} + 324\,n^{3} x\right)\,P_{n}(x) = \left(333 + 1827\,x + 2943\,x^{2} + 2025\,x^{3} + 486\,x^{4} - 414\,n - 1476\,n x - 2214\,n x^{2} - 1539\,n x^{3} - 486\,n x^{4} - 567\,n^{2} - 1296\,n^{2} x - 486\,n^{2} x^{2} + 81\,n^{3} + 567\,n^{3} x + 486\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-340 - 2184\,x - 4923\,x^{2} - 5076\,x^{3} - 2511\,x^{4} - 486\,x^{5} - 180\,n - 918\,n x - 1296\,n x^{2} - 486\,n x^{3} + 45\,n^{2} + 432\,n^{2} x + 1053\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
2356318116
312036929754840
43881701238512151625851
5120968971401312447461748639669
6367125812690938926257591167671458263654
710935914343044795204524884302466455904943741725798

All 15 rows

Technical details
Node id
n-656456ed95c24d09 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f000abd28dc7ac38

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:     9,       6;
  2:    35,      63,      18;
  3:   120,     369,     297,       54;
  4:   388,    1701,    2385,     1215,      162;
  5:  1209,    6897,   14013,    12447,     4617,      486;
  6:  3671,   25812,   69093,    89262,    57591,    16767,     1458;
  7: 10935,   91434,  304479,   520452,   488430,   246645,    59049,    4374;
  8: 32095,  311139, 1240875,  2649267,  3293460,  2416635,  1000917,  203391,  13122;
  9: 93102, 1026801, 4774005, 12250035, 19051686, 18517329, 11129643, 3903795, 688905, 39366;
%F Formula
The row polynomials satisfy (54*n^3-405*n^2+267*n+21+(324*n^3-972*n^2+576*n+72)*x-(378*n-459)*x^2-(324*n-324)*x^3)*P(n,x) = (81*n^3-567*n^2-414*n+333+(567*n^3-1296*n^2-1476*n+1827)*x+(486*n^3-486*n^2-2214*n+2943)*x^2-(1539*n-2025)*x^3-(486*n-486)*x^4)*P(n-1,x) + (45*n^2-180*n-340+(432*n^2-918*n-2184)*x+(1053*n^2-1296*n-4923)*x^2+(486*n^2-486*n-5076)*x^3-2511*x^4-486*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[((81*n^3 - 567*n^2 - 414*n + 333 + (567*n^3 - 1296*n^2 - 1476*n + 1827)*x + (486*n^3 - 486*n^2 - 2214*n + 2943)*x^2 - (1539*n - 2025)*x^3 - (486*n - 486)*x^4)*P[n - 1, x] + (45*n^2 - 180*n - 340 + (432*n^2 - 918*n - 2184)*x + (1053*n^2 - 1296*n - 4923)*x^2 + (486*n^2 - 486*n - 5076)*x^3 - 2511*x^4 - 486*x^5)*D[P[n - 1, x], x])/(54*n^3 - 405*n^2 + 267*n + 21 + (324*n^3 - 972*n^2 + 576*n + 72)*x - (378*n - 459)*x^2 - (324*n - 324)*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] = ((81*n^3 - 567*n^2 - 414*n + 333 + (567*n^3 - 1296*n^2 - 1476*n + 1827)*x + (486*n^3 - 486*n^2 - 2214*n + 2943)*x^2 - (1539*n - 2025)*x^3 - (486*n - 486)*x^4)*v[i - 1] + (45*n^2 - 180*n - 340 + (432*n^2 - 918*n - 2184)*x + (1053*n^2 - 1296*n - 4923)*x^2 + (486*n^2 - 486*n - 5076)*x^3 - 2511*x^4 - 486*x^5)*deriv(v[i - 1], x))/(54*n^3 - 405*n^2 + 267*n + 21 + (324*n^3 - 972*n^2 + 576*n + 72)*x - (378*n - 459)*x^2 - (324*n - 324)*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023