Real-rooted atlas

A114656 → basis A013609 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-7 - 6\,x + 11\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-3 - 8\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2527
33124459
4215246848553
5159724161256256165541
612425233701636052007203258107
7999552247601980608608018960192064629799
88246752157470229334012681203869606339249281286999013

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 + n\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} - n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-5 - 2\,x + 10\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)
Technical details
Node id
n-3059d210b3bacecf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-9db5e9289aa03213)
Fingerprint
f6-1aca77fc72184605

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        5,         2;
   3:       31,        24,         4;
   4:      215,       246,        84,         8;
   5:     1597,      2416,      1256,       256,        16;
   6:    12425,     23370,     16360,      5200,       720,       32;
   7:    99955,    224760,    198060,     86080,     18960,     1920,      64;
   8:   824675,   2157470,   2293340,   1268120,    386960,    63392,    4928,    128;
   9:  6939769,  20705184,  25780944,  17332224,   6791904,  1559040,  198912,  12288,   256;
  10: 59334605, 198823122, 283837392, 224878752, 107890272, 32066496, 5789952, 594432, 29952, 512;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (11*n-7+(6*n-6)*x)*P(n-1,x) - (3+8*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((11*n - 7 + (6*n - 6)*x)*P[n - 1, x] - (3 + 8*x + 4*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((11*n - 7 + (6*n - 6)*x)*v[i - 1] - (3 + 8*x + 4*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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