Real-rooted atlas

A062264 → x ↦ x+1 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-4\,n + 8\,n x + 4\,n x^{2} + 15\,n^{2} + 10\,n^{2} x + n^{2} x^{2} + 8\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(24 + 60\,x + 48\,x^{2} + 12\,x^{3} + 76\,n + 130\,n x + 68\,n x^{2} + 10\,n x^{3} + 82\,n^{2} + 95\,n^{2} x + 33\,n^{2} x^{2} + 3\,n^{2} x^{3} + 34\,n^{3} + 32\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-12\,x - 28\,x^{2} - 20\,x^{3} - 4\,x^{4} - 26\,n x - 42\,n x^{2} - 18\,n x^{3} - 2\,n x^{4} - 15\,n^{2} x - 19\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111516
2707215157
3372588273351268
4181538402772784709281
5843722275217809660189012664168
63803812012014800589760277204032210427885
71679606166169159157057053003006930078543302783980

All 15 rows

Technical details
Node id
n-65efdc6766f19c5b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9e622d62f62e52cc

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:      11,        5;
  2:      70,       72,       15;
  3:     372,      588,      273,       35;
  4:    1815,     3840,     2772,      784,       70;
  5:    8437,    22275,    21780,     9660,     1890,     126;
  6:   38038,   120120,   148005,    89760,    27720,    4032,     210;
  7:  167960,   616616,   915915,   705705,   300300,   69300,    7854,    330;
  8:  730626,  3055104,  5309304,  4964960,  2702700,  864864,  156156,  14256,   495;
  9: 3142790, 14738490, 29325816, 32264232, 21441420, 8828820, 2222220, 324324, 24453, 715;
%F Formula
The row polynomials satisfy (n^4+8*n^3+15*n^2-4*n+(2*n^3+10*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (4*n^4+34*n^3+82*n^2+76*n+24+(3*n^4+32*n^3+95*n^2+130*n+60)*x+(6*n^3+33*n^2+68*n+48)*x^2+(3*n^2+10*n+12)*x^3)*P(n-1,x) - ((2*n^3+15*n^2+26*n+12)*x+(2*n^3+19*n^2+42*n+28)*x^2+(4*n^2+18*n+20)*x^3+(2*n+4)*x^4)*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[((4*n^4 + 34*n^3 + 82*n^2 + 76*n + 24 + (3*n^4 + 32*n^3 + 95*n^2 + 130*n + 60)*x + (6*n^3 + 33*n^2 + 68*n + 48)*x^2 + (3*n^2 + 10*n + 12)*x^3)*P[n - 1, x] - ((2*n^3 + 15*n^2 + 26*n + 12)*x + (2*n^3 + 19*n^2 + 42*n + 28)*x^2 + (4*n^2 + 18*n + 20)*x^3 + (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 8*n^3 + 15*n^2 - 4*n + (2*n^3 + 10*n^2 + 8*n)*x + (n^2 + 4*n)*x^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] = ((4*n^4 + 34*n^3 + 82*n^2 + 76*n + 24 + (3*n^4 + 32*n^3 + 95*n^2 + 130*n + 60)*x + (6*n^3 + 33*n^2 + 68*n + 48)*x^2 + (3*n^2 + 10*n + 12)*x^3)*v[i - 1] - ((2*n^3 + 15*n^2 + 26*n + 12)*x + (2*n^3 + 19*n^2 + 42*n + 28)*x^2 + (4*n^2 + 18*n + 20)*x^3 + (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 8*n^3 + 15*n^2 - 4*n + (2*n^3 + 10*n^2 + 8*n)*x + (n^2 + 4*n)*x^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. [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.