Real-rooted atlas

A126216 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 3\,n\right)\,P_{n}(x) = \left(6 + 48\,n + 27\,n x\right)\,P_{n-1}(x) + \left(-40 - 54\,x - 18\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
24139989
330642318927945
42426438028808108110577
5200764467038700162003240243123129
617148145321348694527067581405123937291476841
71500666459183958875394081455164146537932345927218718130401
81338620646533288693231845758981229031534902890816737841662126561226739489
9121267476471993210801086760775746720470414952184228506462331807085880590490196832878666857
10111267402647932317909129541050100980513007159100760338443207210740794402240581502903242520667155904937006326777
1110318939956487397989981029417948901280469019501039367716205760278647222138406136586317204010392624001158344557144929177147480750990993
1296572168916496252543464115102764374415904060068601454594396310925021865592417835063008134414734608303594528604650699240452079144244462865314416301611631473
139108969928565059024107228127831068041041942552479067219767739799808141902065124827370046900216279314241148876844967170415051723944420198064744173178907282904796159432383240669582937
148650566601401516354492058651411782621262652340115614796052622609274415132097858620787931232020042303055379065607415517473422931965417994390113371952228047785518599985652875268527900652547829691106980509493641

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(16 + 24\,x + 9\,x^{2} - 16\,n - 24\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x + 12\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)
Technical details
Node id
n-e1a2b20716b1e97c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-828cb1728507cda8)
Fingerprint
f6-08e5ad0dcb445c93

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:         6,         3;
  2:        41,        39,         9;
  3:       306,       423,       189,        27;
  4:      2426,      4380,      2880,       810,        81;
  5:     20076,     44670,     38700,     16200,      3240,       243;
  6:    171481,    453213,    486945,    270675,     81405,     12393,      729;
  7:   1500666,   4591839,   5887539,   4081455,   1641465,    379323,    45927,    2187;
  8:  13386206,  46533288,  69323184,  57589812,  29031534,   9028908,  1673784,  166212,   6561;
  9: 121267476, 471993210, 801086760, 775746720, 470414952, 184228506, 46233180, 7085880, 590490, 19683;
%F Formula
The row polynomials satisfy (3*n+6)*P(n,x) = (48*n+6+27*n*x)*P(n-1,x) - (40+54*x+18*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[((48*n + 6 + 27*n*x)*P[n - 1, x] - (40 + 54*x + 18*x^2)*D[P[n - 1, x], x])/(3*n + 6)]];
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] = ((48*n + 6 + 27*n*x)*v[i - 1] - (40 + 54*x + 18*x^2)*deriv(v[i - 1], x))/(3*n + 6)); 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