Real-rooted atlas

A001263 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 13\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-8 - 14\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
22933971
318530616227680
4125727182124702816882
58925238052457012150283524372528
6654452074952660851753656196510935729788019
749182518064202768220228123010829702925724082421878766248
837682091573105228039536277436881656320460555601306368148716656199362894
929324405137122983278688924321906312231737436107101764314753045589972531441196831143498224
10231153133119672380527313522653607808850304210242017020137966353235001546427702312752518698855904913326176998
1118418010651045794941426482778190393757447053806916003025024091400113487454083524109930726401115931005906495390177147156950554384
12148100694979150962711425460386657242083714079045899340646534679926472418530972380870147613844185216045853288832470366339996223205225314411865210341828
131200296578058017418481932431055343366442224054300653716765395154587883053039282434770846812630320497084078033180659311206518914439251970141398735475996063159432322338852956064
149794701406617032685348899230824926447574582530106163161353828348813584600615669074075885932218121044539855053804702889890922533761520274509353058216175716428753680855412566860034782969269355965364459

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(9 + 18\,x + 9\,x^{2} - 9\,n - 18\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(5 + 3\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)
Technical details
Node id
n-97fcfd2b195b6955 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-efa87cd8b4b765e4)
Fingerprint
f6-b59858f0c3c98e59

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:        5,         3;
  2:       29,        33,         9;
  3:      185,       306,       162,        27;
  4:     1257,      2718,      2124,       702,        81;
  5:     8925,     23805,     24570,     12150,      2835,       243;
  6:    65445,    207495,    266085,    175365,     61965,     10935,      729;
  7:   491825,   1806420,   2768220,   2281230,   1082970,    292572,    40824,    2187;
  8:  3768209,  15731052,  28039536,  27743688,  16563204,   6055560,  1306368,  148716,   6561;
  9: 29324405, 137122983, 278688924, 321906312, 231737436, 107101764, 31475304, 5589972, 531441, 19683;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (13*n+2+9*n*x)*P(n-1,x) - (8+14*x+6*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[((13*n + 2 + 9*n*x)*P[n - 1, x] - (8 + 14*x + 6*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] = ((13*n + 2 + 9*n*x)*v[i - 1] - (8 + 14*x + 6*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

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