Real-rooted atlas

A008957 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-1 - 2\,x - 2\,n - n^{2} + 4\,n^{2} x - n^{2} x^{2} - 2\,n^{4} x + 2\,n^{4} x^{2} - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} - 6\,n x^{2} - 3\,n^{2} x^{3} + 2\,n^{3} x^{2} - 4\,n^{3} x^{3} + n^{4} x^{3} + 2\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3} - 2\,n x^{3} - n^{2} x^{3} - n^{2} x^{4} - 2\,n^{3} x^{4} - n^{4} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 2\,n + 2\,n^{2} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(2\,n^{2} x^{2} - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x) + \left(n^{2} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2110314
312863496
41604413405847
5111018815632170569335
611826006457606506581907125211
71280158342459609802657108923822781991467

All 15 rows

Technical details
Node id
n-406a49c8091873e9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2fe60709b4450df1

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,   2;
  2: 1,  10,     3;
  3: 1,  28,    63,       4;
  4: 1,  60,   441,     340,        5;
  5: 1, 110,  1881,    5632,     1705,         6;
  6: 1, 182,  6006,   45760,    65065,      8190,         7;
  7: 1, 280, 15834,  245960,   980265,    710892,     38227,        8;
  8: 1, 408, 36414, 1005992,  8666515,  19531980,   7502593,   174760,      9;
  9: 1, 570, 75582, 3385040, 53936155, 279527430, 372099553, 77344288, 786429, 10;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(n^2+2*n+1+(2*n^4-4*n^2+2)*x+(n^6-2*n^4+n^2)*x^2)*P(n-2,x) + ((2*n^3-6*n-4)*x^2+(2*n^5+n^4-4*n^3-3*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+2*n+1)*x^3+(n^4+2*n^3+n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+2*n+1+2*n^4*x)*P(n-1,x) - (2*n^3-2*n^2)*x^2*d/dx P(n-1,x) + n^2*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 2*n + 1 + (2*n^4 - 4*n^2 + 2)*x + (n^6 - 2*n^4 + n^2)*x^2)*P[n - 2, x] + ((2*n^3 - 6*n - 4)*x^2 + (2*n^5 + n^4 - 4*n^3 - 3*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + 2*n + 1)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 2*n + 1 + 2*n^4*x)*P[n - 1, x] - (2*n^3 - 2*n^2)*x^2*D[P[n - 1, x], x] + n^2*x^3*D[P[n - 1, x], {x, 2}])/(n^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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 2*n + 1 + (2*n^4 - 4*n^2 + 2)*x + (n^6 - 2*n^4 + n^2)*x^2)*v[i - 2] + ((2*n^3 - 6*n - 4)*x^2 + (2*n^5 + n^4 - 4*n^3 - 3*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + 2*n + 1)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 2*n + 1 + 2*n^4*x)*v[i - 1] - (2*n^3 - 2*n^2)*x^2*deriv(v[i - 1], x) + n^2*x^3*deriv(deriv(v[i - 1], x), x))/(n^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.