Real-rooted atlas

A152198 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-78 - 126\,x + 108\,x^{2} - 9\,n + 108\,n x + 27\,n^{2}\right)\,P_{n}(x) = \left(72 - 36\,x + 54\,x^{2} - 324\,x^{3} - 204\,n - 63\,n x - 270\,n x^{2} + 324\,n x^{3} + 45\,n^{2} + 324\,n^{2} x^{2} + 27\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(-40 - 372\,x - 972\,x^{2} - 1188\,x^{3} - 648\,x^{4} - 132\,n - 522\,n x - 918\,n x^{2} - 648\,n x^{3} - 36\,n^{2} - 162\,n^{2} x - 162\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2538
3538
41630955
51630955
64414413527350
74414413527350
8112528864540812125
9112528864540812125
10272168039604320202524312500
11272168039604320202524312500
126404896151202376019440729072971875
136404896151202376019440729072971875
141472134405140810584012474081648255152187406250
Technical details
Node id
n-f97899e6ddb309b6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1e01e0fee2e89b30

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   1;
  2:   5,   3;
  3:   5,   3;
  4:  16,  30,   9;
  5:  16,  30,   9;
  6:  44, 144, 135,  27;
  7:  44, 144, 135,  27;
  8: 112, 528, 864, 540, 81;
  9: 112, 528, 864, 540, 81;
%F Formula
The row polynomials satisfy (27*n^2-9*n-78+(108*n-126)*x+108*x^2)*P(n,x) = (27*n^3+45*n^2-204*n+72+(81*n^3-63*n-36)*x+(324*n^2-270*n+54)*x^2+(324*n-324)*x^3)*P(n-1,x) - (36*n^2+132*n+40+(162*n^2+522*n+372)*x+(162*n^2+918*n+972)*x^2+(648*n+1188)*x^3+648*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[((27*n^3 + 45*n^2 - 204*n + 72 + (81*n^3 - 63*n - 36)*x + (324*n^2 - 270*n + 54)*x^2 + (324*n - 324)*x^3)*P[n - 1, x] - (36*n^2 + 132*n + 40 + (162*n^2 + 522*n + 372)*x + (162*n^2 + 918*n + 972)*x^2 + (648*n + 1188)*x^3 + 648*x^4)*D[P[n - 1, x], x])/(27*n^2 - 9*n - 78 + (108*n - 126)*x + 108*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] = ((27*n^3 + 45*n^2 - 204*n + 72 + (81*n^3 - 63*n - 36)*x + (324*n^2 - 270*n + 54)*x^2 + (324*n - 324)*x^3)*v[i - 1] - (36*n^2 + 132*n + 40 + (162*n^2 + 522*n + 372)*x + (162*n^2 + 918*n + 972)*x^2 + (648*n + 1188)*x^3 + 648*x^4)*deriv(v[i - 1], x))/(27*n^2 - 9*n - 78 + (108*n - 126)*x + 108*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. [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.