Real-rooted atlas

A113214 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n^{2} x + n^{3} - 6\,n^{3} x + n^{3} x^{2} + 2\,n^{4} - 3\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(-4\,x - 8\,x^{2} + 2\,x^{3} - 14\,n x - 28\,n x^{2} + 4\,n x^{3} + 4\,n^{2} - 31\,n^{2} x^{2} + 2\,n^{2} x^{3} + 6\,n^{3} + 20\,n^{3} x - 11\,n^{3} x^{2} + 10\,n^{4} x - 2\,n^{5}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 2\,x^{3} + x^{4} - 32\,n x^{2} + 2\,n x^{4} + 4\,n^{2} x - 35\,n^{2} x^{2} - 5\,n^{2} x^{3} + n^{2} x^{4} + 8\,n^{3} x - 10\,n^{3} x^{2} - 3\,n^{3} x^{3} + 4\,n^{4} x + n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
20242751
301610864188
400135320125580
500545767502161596
6000448175015123434053
70001282000432027445129704
80000112564809261460872922203
90000250540017150179207290100049010
10000002376188653942432076110001331105072
110000043212348537608164854000159721728219888
1200000044594659213267815600086515224642197450905
1300000068625088142884294000279510133056307582744908726
14000000076801020603780005989504752001977304116033751804155
Technical details
Node id
n-78d3743b85208d75 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2be0a5b069dfbb51

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: 2,  8;
  2: 0, 24,  27;
  3: 0, 16, 108,  64;
  4: 0,  0, 135, 320,  125;
  5: 0,  0,  54, 576,  750,  216;
  6: 0,  0,   0, 448, 1750, 1512,   343;
  7: 0,  0,   0, 128, 2000, 4320,  2744,   512;
  8: 0,  0,   0,   0, 1125, 6480,  9261,  4608,  729;
  9: 0,  0,   0,   0,  250, 5400, 17150, 17920, 7290, 1000;
%F Formula
The row polynomials satisfy (n^5+2*n^4+n^3-(3*n^4+6*n^3+n^2)*x+n^3*x^2)*P(n,x) = -(2*n^5-6*n^3-4*n^2-(10*n^4+20*n^3-14*n-4)*x+(11*n^3+31*n^2+28*n+8)*x^2-(2*n^2+4*n+2)*x^3)*P(n-1,x) + ((4*n^4+8*n^3+4*n^2)*x+(n^4-10*n^3-35*n^2-32*n-8)*x^2-(3*n^3+5*n^2-2)*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[(-(2*n^5 - 6*n^3 - 4*n^2 - (10*n^4 + 20*n^3 - 14*n - 4)*x + (11*n^3 + 31*n^2 + 28*n + 8)*x^2 - (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + ((4*n^4 + 8*n^3 + 4*n^2)*x + (n^4 - 10*n^3 - 35*n^2 - 32*n - 8)*x^2 - (3*n^3 + 5*n^2 - 2)*x^3 + (n^2 + 2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^5 + 2*n^4 + n^3 - (3*n^4 + 6*n^3 + n^2)*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] = (-(2*n^5 - 6*n^3 - 4*n^2 - (10*n^4 + 20*n^3 - 14*n - 4)*x + (11*n^3 + 31*n^2 + 28*n + 8)*x^2 - (2*n^2 + 4*n + 2)*x^3)*v[i - 1] + ((4*n^4 + 8*n^3 + 4*n^2)*x + (n^4 - 10*n^3 - 35*n^2 - 32*n - 8)*x^2 - (3*n^3 + 5*n^2 - 2)*x^3 + (n^2 + 2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^5 + 2*n^4 + n^3 - (3*n^4 + 6*n^3 + n^2)*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

Same fingerprint ?

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.