Real-rooted atlas

row-factor cofactor of A101842 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4\,x - 7\,n x^{2} - 4\,n^{2} x - 2\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 5\,x^{2} - 2\,n x + 8\,n x^{2} - 13\,n x^{3} + 4\,n^{2} x^{2} - 14\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + x^{3} - n x^{2} + 4\,n x^{3} - 3\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(n + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 2\,n x - 4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
2116926
31449916160
41104594416251140
51228271848321425369240
6148010719386562977543204984000
71988386372499043904751545481210364846720

All 15 rows

Technical details
Node id
n-dd7b6b4ebf46d443 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3b4269c91cbc7ccc

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: 1,    4;
  2: 1,   16,      9;
  3: 1,   44,     99,      16;
  4: 1,  104,    594,     416,       25;
  5: 1,  228,   2718,    4832,     1425,       36;
  6: 1,  480,  10719,   38656,    29775,     4320,       49;
  7: 1,  988,  38637,  249904,   390475,   154548,    12103,      64;
  8: 1, 2008, 131472, 1411744,  3904750,  3176424,   715792,   32128,    81;
  9: 1, 4052, 430560, 7283072, 32758850, 47172744, 22304408, 3061760, 82053, 100;
%F Formula
The row polynomials satisfy n*P(n,x) = -((4*n^2-4)*x-(9*n^3-2*n^2-7*n)*x^2)*P(n-2,x) - ((2*n+2)*x-(4*n^2+8*n+5)*x^2-(4*n^3-14*n^2-13*n)*x^3)*d/dx P(n-2,x) - ((n+1)*x^2-(4*n^2+4*n+1)*x^3+(4*n^2+3*n)*x^4)*d^2/dx^2 P(n-2,x) + (n+4*n^2*x)*P(n-1,x) + ((2*n+1)*x-(4*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 - 4)*x - (9*n^3 - 2*n^2 - 7*n)*x^2)*P[n - 2, x] - ((2*n + 2)*x - (4*n^2 + 8*n + 5)*x^2 - (4*n^3 - 14*n^2 - 13*n)*x^3)*D[P[n - 2, x], x] - ((n + 1)*x^2 - (4*n^2 + 4*n + 1)*x^3 + (4*n^2 + 3*n)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 4*n^2*x)*P[n - 1, x] + ((2*n + 1)*x - (4*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^2 - 4)*x - (9*n^3 - 2*n^2 - 7*n)*x^2)*v[i - 2] - ((2*n + 2)*x - (4*n^2 + 8*n + 5)*x^2 - (4*n^3 - 14*n^2 - 13*n)*x^3)*deriv(v[i - 2], x) - ((n + 1)*x^2 - (4*n^2 + 4*n + 1)*x^3 + (4*n^2 + 3*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 4*n^2*x)*v[i - 1] + ((2*n + 1)*x - (4*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.