Real-rooted atlas

A133087 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + x^{2} + n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-10 + 2\,x + 11\,x^{2} + 3\,x^{3} + 11\,n + 17\,n x - n x^{2} - 2\,n x^{3} + 4\,n^{2} - 9\,n^{2} x - 4\,n^{2} x^{2} - 5\,n^{3} - 2\,n^{3} x\right)\,P_{n-1}(x) + \left(4 + 22\,x + 28\,x^{2} + 13\,x^{3} + 2\,x^{4} + 18\,n + 37\,n x + 22\,n x^{2} + 4\,n x^{3} + 10\,n^{2} + 9\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
286115
3283010169
4488048121189
516032024892161837
62566727204001201812187
78322464307220808201862219477
812804608716862723360112022424124057
940961612827904277761747271681904312281102789
1061442816057600691205376028224100802400360301255879
111945695744212480280320243840146496618241824036604703411082565
1228672159744405504619520633600456192236544887042376044005283612657205
13901125324801443840237670426470402103552122496052800016790438720624866040111160261
1413107286016026091524845568615014456376323843840197683276876822422448048728072842127103491
Technical details
Node id
n-bd06d5df5d01b1a2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ec3a2d9737ae1408

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:    4,     1;
  2:    8,     6,     1;
  3:   28,    30,    10,     1;
  4:   48,    80,    48,    12,     1;
  5:  160,   320,   248,    92,    16,    1;
  6:  256,   672,   720,   400,   120,   18,    1;
  7:  832,  2464,  3072,  2080,   820,  186,   22,   1;
  8: 1280,  4608,  7168,  6272,  3360, 1120,  224,  24,  1;
  9: 4096, 16128, 27904, 27776, 17472, 7168, 1904, 312, 28, 1;
%F Formula
The row polynomials satisfy (n^2+n-2+2*n*x+x^2)*P(n,x) = -(5*n^3-4*n^2-11*n+10+(2*n^3+9*n^2-17*n-2)*x+(4*n^2+n-11)*x^2+(2*n-3)*x^3)*P(n-1,x) + (10*n^2+18*n+4+(9*n^2+37*n+22)*x+(2*n^2+22*n+28)*x^2+(4*n+13)*x^3+2*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[(-(5*n^3 - 4*n^2 - 11*n + 10 + (2*n^3 + 9*n^2 - 17*n - 2)*x + (4*n^2 + n - 11)*x^2 + (2*n - 3)*x^3)*P[n - 1, x] + (10*n^2 + 18*n + 4 + (9*n^2 + 37*n + 22)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n + 13)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 + 2*n*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; for(n = 1, N, my(i = n + 1); v[i] = (-(5*n^3 - 4*n^2 - 11*n + 10 + (2*n^3 + 9*n^2 - 17*n - 2)*x + (4*n^2 + n - 11)*x^2 + (2*n - 3)*x^3)*v[i - 1] + (10*n^2 + 18*n + 4 + (9*n^2 + 37*n + 22)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n + 13)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 + 2*n*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.