Real-rooted atlas

A274496 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-26 - 42\,x + 36\,x^{2} - 3\,n + 36\,n x + 9\,n^{2}\right)\,P_{n}(x) = \left(8 - 96\,x + 90\,x^{2} - 108\,x^{3} - 102\,n + 93\,n x - 126\,n x^{2} + 108\,n x^{3} + 54\,n^{2} - 36\,n^{2} x + 108\,n^{2} x^{2} + 27\,n^{3} x\right)\,P_{n-1}(x) + \left(-16 - 136\,x - 288\,x^{2} - 324\,x^{3} - 216\,x^{4} - 48\,n - 132\,n x - 234\,n x^{2} - 216\,n x^{3} - 36\,n^{2} x - 54\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
210616
3201232
46412036220
512824072440
6352115210802162800
7704230421604325600

All 15 rows

Technical details
Node id
n-00447aedf6d1c69b · 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:    2;
  2:   10,     6;
  3:   20,    12;
  4:   64,   120,    36;
  5:  128,   240,    72;
  6:  352,  1152,  1080,   216;
  7:  704,  2304,  2160,   432;
  8: 1792,  8448, 13824,  8640, 1296;
  9: 3584, 16896, 27648, 17280, 2592;
%F Formula
The row polynomials satisfy (9*n^2-3*n-26+(36*n-42)*x+36*x^2)*P(n,x) = (54*n^2-102*n+8+(27*n^3-36*n^2+93*n-96)*x+(108*n^2-126*n+90)*x^2+(108*n-108)*x^3)*P(n-1,x) - (48*n+16+(36*n^2+132*n+136)*x+(54*n^2+234*n+288)*x^2+(216*n+324)*x^3+216*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[((54*n^2 - 102*n + 8 + (27*n^3 - 36*n^2 + 93*n - 96)*x + (108*n^2 - 126*n + 90)*x^2 + (108*n - 108)*x^3)*P[n - 1, x] - (48*n + 16 + (36*n^2 + 132*n + 136)*x + (54*n^2 + 234*n + 288)*x^2 + (216*n + 324)*x^3 + 216*x^4)*D[P[n - 1, x], x])/(9*n^2 - 3*n - 26 + (36*n - 42)*x + 36*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] = ((54*n^2 - 102*n + 8 + (27*n^3 - 36*n^2 + 93*n - 96)*x + (108*n^2 - 126*n + 90)*x^2 + (108*n - 108)*x^3)*v[i - 1] - (48*n + 16 + (36*n^2 + 132*n + 136)*x + (54*n^2 + 234*n + 288)*x^2 + (216*n + 324)*x^3 + 216*x^4)*deriv(v[i - 1], x))/(9*n^2 - 3*n - 26 + (36*n - 42)*x + 36*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.