Real-rooted atlas

A145596 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + n^{2}\right)\,P_{n}(x) = \left(8 + 2\,x + 15\,n + 5\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2} - 4\,n - 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
23120354
3156148444352
47859804358052285
53978615036301010130614904
6203353746427594103042030196797930
7104856224168197904928482475236962808648512

All 15 rows

Technical details
Node id
n-def8bd320e07d51b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e0fd39fe6c3470da

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,       2;
  2:      31,      20,       3;
  3:     156,     148,      44,       4;
  4:     785,     980,     435,      80,       5;
  5:    3978,    6150,    3630,    1010,     130,      6;
  6:   20335,   37464,   27594,   10304,    2030,    196,      7;
  7:  104856,  224168,  197904,   92848,   24752,   3696,    280,     8;
  8:  545073, 1326072, 1364706,  771456,  259980,  52920,   6258,   384,   9;
  9: 2854350, 7784970, 9150780, 6055140, 2470440, 639240, 103740, 10020, 510, 10;
%F Formula
The row polynomials satisfy (n^2+4*n)*P(n,x) = (7*n^2+15*n+8+(3*n^2+5*n+2)*x)*P(n-1,x) - (4*n+4+(6*n+6)*x+(2*n+2)*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[((7*n^2 + 15*n + 8 + (3*n^2 + 5*n + 2)*x)*P[n - 1, x] - (4*n + 4 + (6*n + 6)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
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] = ((7*n^2 + 15*n + 8 + (3*n^2 + 5*n + 2)*x)*v[i - 1] - (4*n + 4 + (6*n + 6)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.