Real-rooted atlas

A114687 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-30 - 162\,x + 108\,x^{2} - 51\,n + 135\,n x + 54\,n x^{2} + 90\,n^{2} + 108\,n^{2} x + 54\,n^{3}\right)\,P_{n}(x) = \left(-30 - 162\,x + 108\,x^{2} - 87\,n - 783\,n x - 108\,n x^{2} + 972\,n x^{3} + 306\,n^{2} + 1080\,n^{2} x + 1944\,n^{2} x^{2} + 378\,n^{3} + 972\,n^{3} x\right)\,P_{n-1}(x) + \left(-8 - 90\,x - 342\,x^{2} - 648\,x^{3} - 648\,x^{4} - 42\,n - 450\,n x - 1404\,n x^{2} - 1296\,n x^{3} - 36\,n^{2} - 324\,n^{2} x - 648\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
25311436203
3297111610802162709
416498940154089072129636365
5915365130164520174960712807776492819
650917449478150714023792401769040536544466566739015
7284049299695212570768265960802966544016547328391910427993692859657

All 15 rows

Technical details
Node id
n-57662c9c77b7ffae · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f6cfd7778cb67b91

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:      53,       114,        36;
  3:     297,      1116,      1080,        216;
  4:    1649,      8940,     15408,       9072,       1296;
  5:    9153,     65130,    164520,     174960,      71280,       7776;
  6:   50917,    449478,   1507140,    2379240,    1769040,     536544,      46656;
  7:  284049,   2996952,  12570768,   26596080,   29665440,   16547328,    3919104,     279936;
  8: 1589153,  19517208,  98465472,  262495296,  395648064,  333994752,  146313216,   27993600,   1679616;
  9: 8914617, 124960590, 737484912, 2377988928, 4545180864, 5222206080, 3488002560, 1239556608, 196515072, 10077696;
%F Formula
The row polynomials satisfy (54*n^3+90*n^2-51*n-30+(108*n^2+135*n-162)*x+(54*n+108)*x^2)*P(n,x) = (378*n^3+306*n^2-87*n-30+(972*n^3+1080*n^2-783*n-162)*x+(1944*n^2-108*n+108)*x^2+972*n*x^3)*P(n-1,x) - (36*n^2+42*n+8+(324*n^2+450*n+90)*x+(648*n^2+1404*n+342)*x^2+(1296*n+648)*x^3+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[((378*n^3 + 306*n^2 - 87*n - 30 + (972*n^3 + 1080*n^2 - 783*n - 162)*x + (1944*n^2 - 108*n + 108)*x^2 + 972*n*x^3)*P[n - 1, x] - (36*n^2 + 42*n + 8 + (324*n^2 + 450*n + 90)*x + (648*n^2 + 1404*n + 342)*x^2 + (1296*n + 648)*x^3 + 648*x^4)*D[P[n - 1, x], x])/(54*n^3 + 90*n^2 - 51*n - 30 + (108*n^2 + 135*n - 162)*x + (54*n + 108)*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] = ((378*n^3 + 306*n^2 - 87*n - 30 + (972*n^3 + 1080*n^2 - 783*n - 162)*x + (1944*n^2 - 108*n + 108)*x^2 + 972*n*x^3)*v[i - 1] - (36*n^2 + 42*n + 8 + (324*n^2 + 450*n + 90)*x + (648*n^2 + 1404*n + 342)*x^2 + (1296*n + 648)*x^3 + 648*x^4)*deriv(v[i - 1], x))/(54*n^3 + 90*n^2 - 51*n - 30 + (108*n^2 + 135*n - 162)*x + (54*n + 108)*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