Real-rooted atlas

A113953 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16\,n + 4\,n x + 48\,n^{2} - 18\,n^{2} x + 52\,n^{3} - 24\,n^{3} x + n^{3} x^{2} + 24\,n^{4} - 6\,n^{4} x + 4\,n^{5}\right)\,P_{n}(x) = \left(64 + 144\,x - 72\,x^{2} + 2\,x^{3} + 128\,n + 352\,n x - 156\,n x^{2} + 4\,n x^{3} + 16\,n^{2} + 400\,n^{2} x - 102\,n^{2} x^{2} + 2\,n^{2} x^{3} - 112\,n^{3} + 216\,n^{3} x - 22\,n^{3} x^{2} - 80\,n^{4} + 40\,n^{4} x - 16\,n^{5}\right)\,P_{n-1}(x) + \left(128\,x - 144\,x^{2} - 12\,x^{3} + x^{4} + 384\,n x - 304\,n x^{2} - 28\,n x^{3} + 2\,n x^{4} + 416\,n^{2} x - 188\,n^{2} x^{2} - 22\,n^{2} x^{3} + n^{2} x^{4} + 192\,n^{3} x - 24\,n^{3} x^{2} - 6\,n^{3} x^{3} + 32\,n^{4} x + 4\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1088
20162743
30010864172
400108384125617
500076810002161984
6000512300021603436015
7000040008640411651217268

All 15 rows

Technical details
Node id
n-6aca4afa0b30a188 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cbd07c4dfcbe14b4

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: 0,  8;
  2: 0, 16,  27;
  3: 0,  0, 108,  64;
  4: 0,  0, 108, 384,  125;
  5: 0,  0,   0, 768, 1000,   216;
  6: 0,  0,   0, 512, 3000,  2160,   343;
  7: 0,  0,   0,   0, 4000,  8640,  4116,   512;
  8: 0,  0,   0,   0, 2000, 17280, 20580,  7168,   729;
  9: 0,  0,   0,   0,    0, 17280, 54880, 43008, 11664, 1000;
%F Formula
The row polynomials satisfy (4*n^5+24*n^4+52*n^3+48*n^2+16*n-(6*n^4+24*n^3+18*n^2-4*n)*x+n^3*x^2)*P(n,x) = -(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64-(40*n^4+216*n^3+400*n^2+352*n+144)*x+(22*n^3+102*n^2+156*n+72)*x^2-(2*n^2+4*n+2)*x^3)*P(n-1,x) + ((32*n^4+192*n^3+416*n^2+384*n+128)*x+(4*n^4-24*n^3-188*n^2-304*n-144)*x^2-(6*n^3+22*n^2+28*n+12)*x^3+(n^2+2*n+1)*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[(-(16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64 - (40*n^4 + 216*n^3 + 400*n^2 + 352*n + 144)*x + (22*n^3 + 102*n^2 + 156*n + 72)*x^2 - (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + ((32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x + (4*n^4 - 24*n^3 - 188*n^2 - 304*n - 144)*x^2 - (6*n^3 + 22*n^2 + 28*n + 12)*x^3 + (n^2 + 2*n + 1)*x^4)*D[P[n - 1, x], x])/(4*n^5 + 24*n^4 + 52*n^3 + 48*n^2 + 16*n - (6*n^4 + 24*n^3 + 18*n^2 - 4*n)*x + n^3*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] = (-(16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64 - (40*n^4 + 216*n^3 + 400*n^2 + 352*n + 144)*x + (22*n^3 + 102*n^2 + 156*n + 72)*x^2 - (2*n^2 + 4*n + 2)*x^3)*v[i - 1] + ((32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x + (4*n^4 - 24*n^3 - 188*n^2 - 304*n - 144)*x^2 - (6*n^3 + 22*n^2 + 28*n + 12)*x^3 + (n^2 + 2*n + 1)*x^4)*deriv(v[i - 1], x))/(4*n^5 + 24*n^4 + 52*n^3 + 48*n^2 + 16*n - (6*n^4 + 24*n^3 + 18*n^2 - 4*n)*x + n^3*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.