Real-rooted atlas

A075497 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-8 - 40\,x - 75\,x^{2} - 66\,x^{3} - 27\,x^{4} - 4\,x^{5} + 4\,n x + 14\,n x^{2} + 18\,n x^{3} + 10\,n x^{4} + 2\,n x^{5}\right)\,P_{n-2}(x) + \left(-4\,x - 18\,x^{2} - 32\,x^{3} - 28\,x^{4} - 12\,x^{5} - 2\,x^{6}\right)\,P_{n-2}'(x) + \left(6 + 20\,x + 24\,x^{2} + 12\,x^{3} + 2\,x^{4} - 2\,n x - 6\,n x^{2} - 6\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 8\,x^{2} + 12\,x^{3} + 8\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
210281957
336160227104527
4136864193618486415425
55284576145242177615679438061463
620802412810265621598424286614029232803760809
78256127744705648196185630705842755712132759126676810224159

All 15 rows

Technical details
Node id
n-7c1476ed9f1e6e34 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5cc2221bc56e6a72

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:      3,       4;
  2:     10,      28,       19;
  3:     36,     160,      227,       104;
  4:    136,     864,     1936,      1848,       641;
  5:    528,    4576,    14524,     21776,     15679,      4380;
  6:   2080,   24128,   102656,    215984,    242866,    140292,     32803;
  7:   8256,  127744,   705648,   1961856,   3070584,   2755712,   1327591,    266768;
  8:  32896,  683008,  4799488,  16989440,  34774496,  43162496,  32167024,  13276464,   2337505;
  9: 131328, 3701248, 32624576, 143394560, 369160784, 594853120, 610142056, 388209568, 140024987, 21925236;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(8-(4*n-40)*x-(14*n-75)*x^2-(18*n-66)*x^3-(10*n-27)*x^4-(2*n-4)*x^5)*P(n-2,x) - (4*x+18*x^2+32*x^3+28*x^4+12*x^5+2*x^6)*d/dx P(n-2,x) + (6-(2*n-20)*x-(6*n-24)*x^2-(6*n-12)*x^3-(2*n-2)*x^4)*P(n-1,x) + (2*x+8*x^2+12*x^3+8*x^4+2*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8 - (4*n - 40)*x - (14*n - 75)*x^2 - (18*n - 66)*x^3 - (10*n - 27)*x^4 - (2*n - 4)*x^5)*P[n - 2, x] - (4*x + 18*x^2 + 32*x^3 + 28*x^4 + 12*x^5 + 2*x^6)*D[P[n - 2, x], x] + (6 - (2*n - 20)*x - (6*n - 24)*x^2 - (6*n - 12)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (2*x + 8*x^2 + 12*x^3 + 8*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8 - (4*n - 40)*x - (14*n - 75)*x^2 - (18*n - 66)*x^3 - (10*n - 27)*x^4 - (2*n - 4)*x^5)*v[i - 2] - (4*x + 18*x^2 + 32*x^3 + 28*x^4 + 12*x^5 + 2*x^6)*deriv(v[i - 2], x) + (6 - (2*n - 20)*x - (6*n - 24)*x^2 - (6*n - 12)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (2*x + 8*x^2 + 12*x^3 + 8*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.