Real-rooted atlas

A069466 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-2\,n - 4\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2} + 8\,n x + 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
26363678
3202406004001260
470140063009800490022470
525275605292014112015876063504424116
6924388083880801552320291060025613288537768305836
7343219219225945921441440039639600570810244122518411778624166929048

All 15 rows

Technical details
Node id
n-d50c685731c311f2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fd15deff59829444

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:     2,       4;
  2:     6,      36,       36;
  3:    20,     240,      600,       400;
  4:    70,    1400,     6300,      9800,       4900;
  5:   252,    7560,    52920,    141120,     158760,       63504;
  6:   924,   38808,   388080,   1552320,    2910600,     2561328,      853776;
  7:  3432,  192192,  2594592,  14414400,   39639600,    57081024,    41225184,    11778624;
  8: 12870,  926640, 16216200, 118918800,  445945500,   927566640,  1082161080,   662547600,   165636900;
  9: 48620, 4375800, 96267600, 898497600, 4380175800, 12264492240, 20440820400, 20023660800, 10637569800, 2363904400;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (4*n^2-2*n+(8*n^2-4*n)*x)*P(n-1,x) + ((8*n-4)*x+(8*n-4)*x^2)*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[((4*n^2 - 2*n + (8*n^2 - 4*n)*x)*P[n - 1, x] + ((8*n - 4)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(n^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] = ((4*n^2 - 2*n + (8*n^2 - 4*n)*x)*v[i - 1] + ((8*n - 4)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(n^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

Candidate equalities ?

Same fingerprint ?

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