Real-rooted atlas

A114687 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 13\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-12 - 14\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22922455
3185204728469
412571812944208164237
58925158701092036005603239907
6654451383301182605196012240144064387739
7491825120428012303206759202139203852835841283858505
83768209104873681246201682203523271744797440114688870425639130777
92932440591415322123861744953796484577529614103936276326432716820736512402972031
10231153133797815870121393434010689804006009091202241335045577600090508809024004864010244202705311
1118418010656971966276117701236401166688732075198340803295353600996323328206257920283430402421760112640204844299426717
12148100694976100641807611315727403212469248616090665364240456690389761626861772841055759367226841608555008063528962580484096471189693925
13120029657805534494565462108046904149613102934942241061071909040604165669536247953708288739223147521591185024024220584962503987201634713658572881925051001609115
1497947014066146884568992661025888561989213577866981224121192747355687698444321568357828120249612316877464323139825328645861548492878199080967139758084137369613189121638454513542257795

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(9 + 12\,x + 4\,x^{2} - 9\,n - 12\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x + 10\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)
Technical details
Node id
n-6148d4afd1c19fd9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-11a7be78b851c6a1)
Fingerprint
f6-14fdad6b6ae97664

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,        2;
  2:       29,       22,         4;
  3:      185,      204,        72,        8;
  4:     1257,     1812,       944,      208,       16;
  5:     8925,    15870,     10920,     3600,      560,       32;
  6:    65445,   138330,    118260,    51960,    12240,     1440,      64;
  7:   491825,  1204280,   1230320,   675920,   213920,    38528,    3584,    128;
  8:  3768209, 10487368,  12462016,  8220352,  3271744,   797440,  114688,   8704,   256;
  9: 29324405, 91415322, 123861744, 95379648, 45775296, 14103936, 2763264, 327168, 20736, 512;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (13*n+2+6*n*x)*P(n-1,x) - (12+14*x+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[((13*n + 2 + 6*n*x)*P[n - 1, x] - (12 + 14*x + 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] = ((13*n + 2 + 6*n*x)*v[i - 1] - (12 + 14*x + 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

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