Real-rooted atlas

A133081 → reverse (shift +1) → (θ+1)²

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 6\,n + 13\,n^{2} - 12\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(3\,x^{2} + 4\,n x^{2} + 12\,n^{2} x - 11\,n^{2} x^{2} - 32\,n^{3} x - 16\,n^{3} x^{2} + 28\,n^{4} x + 36\,n^{4} x^{2} - 8\,n^{5} x - 16\,n^{5} x^{2}\right)\,P_{n-1}(x) + \left(x^{3} + 2\,n x^{3} + 4\,n^{2} x^{2} - 2\,n^{2} x^{3} - 8\,n^{3} x^{2} - 8\,n^{3} x^{3} + 4\,n^{4} x^{2} + 8\,n^{4} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
104913
200002525
300000364985
4000000008181
5000000000100121221
6000000000000169169
70000000000000196225421
80000000000000000289289
900000000000000000324361685
1000000000000000000000441441
110000000000000000000004845291013
12000000000000000000000000625625
1300000000000000000000000006767291405
140000000000000000000000000000841841
Technical details
Node id
n-79376d53a0fd927d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-70a219431d8e5b15

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 4, 9;
  2: 0, 0, 0, 0, 25;
  3: 0, 0, 0, 0,  0, 36, 49;
  4: 0, 0, 0, 0,  0,  0,  0, 0, 81;
  5: 0, 0, 0, 0,  0,  0,  0, 0,  0, 100, 121;
  6: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0, 169;
  7: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0,   0, 196, 225;
  8: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0,   0,   0,   0, 0, 289;
  9: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0,   0,   0,   0, 0,   0, 324, 361;
%F Formula
The row polynomials satisfy (4*n^4-12*n^3+13*n^2-6*n+1)*P(n,x) = -((8*n^5-28*n^4+32*n^3-12*n^2)*x+(16*n^5-36*n^4+16*n^3+11*n^2-4*n-3)*x^2)*P(n-1,x) + ((4*n^4-8*n^3+4*n^2)*x^2+(8*n^4-8*n^3-2*n^2+2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x + 9*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^5 - 28*n^4 + 32*n^3 - 12*n^2)*x + (16*n^5 - 36*n^4 + 16*n^3 + 11*n^2 - 4*n - 3)*x^2)*P[n - 1, x] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (8*n^4 - 8*n^3 - 2*n^2 + 2*n + 1)*x^3)*D[P[n - 1, x], x])/(4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)]];
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] = 4*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^5 - 28*n^4 + 32*n^3 - 12*n^2)*x + (16*n^5 - 36*n^4 + 16*n^3 + 11*n^2 - 4*n - 3)*x^2)*v[i - 1] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (8*n^4 - 8*n^3 - 2*n^2 + 2*n + 1)*x^3)*deriv(v[i - 1], x))/(4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)); 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.