Real-rooted atlas

A114656 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2415
32010131
411284181215
56726722242811597
642245280240048040112425
7274564118423760660090054199955
818304032032022422480080154001540701824675
91244672248934420500488968962242243203224648816939769
108599552193489921833062495047682935296550368611523744108159334605
116019686415049216016124160096744960356428808316672122304010920054601301513972967
1242600857611715235841400734720955046400409305600114605568211115522513280184800770015414501041935
13304291840091287552001204995686492048281604502361600147350016032744448049116672484704029920010560180139784038517
14219090124807120429056010285064192087027466240478651064321794941491246824560648513556481064194568868288466752141442081354455513105
1515884034048055594119168087225255936080994880512049496871424020940984064062822952192134620611842048574528217273056155195047054321856423813179928556219

Other recurrences

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

\(\left(1 + n\right)\,P_{n}(x) = \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}(x) + \left(-4 - x + 8\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + x\)
Technical details
Node id
n-b4f19b398a718e0e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-43ded58dbe73c25e)
Fingerprint
f6-77e051dbf841d62e

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:       4,        1;
   3:      20,       10,        1;
   4:     112,       84,       18,       1;
   5:     672,      672,      224,      28,       1;
   6:    4224,     5280,     2400,     480,      40,      1;
   7:   27456,    41184,    23760,    6600,     900,     54,     1;
   8:  183040,   320320,   224224,   80080,   15400,   1540,    70,    1;
   9: 1244672,  2489344,  2050048,  896896,  224224,  32032,  2464,   88,   1;
  10: 8599552, 19348992, 18330624, 9504768, 2935296, 550368, 61152, 3744, 108, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (8*n-4+(3*n-3)*x)*P(n-1,x) - (4*x+2*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[((8*n - 4 + (3*n - 3)*x)*P[n - 1, x] - (4*x + 2*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] = ((8*n - 4 + (3*n - 3)*x)*v[i - 1] - (4*x + 2*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

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