Real-rooted atlas

A103633 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(14 + 22\,x + 8\,x^{2} - 23\,n - 34\,n x - 12\,n x^{2} + 9\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} - 112\,n - 243\,n x - 178\,n x^{2} - 44\,n x^{3} + 111\,n^{2} + 227\,n^{2} x + 156\,n^{2} x^{2} + 36\,n^{2} x^{3} - 27\,n^{3} - 54\,n^{3} x - 36\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(8 + 28\,x + 36\,x^{2} + 20\,x^{3} + 4\,x^{4} - 60\,n - 178\,n x - 194\,n x^{2} - 92\,n x^{3} - 16\,n x^{4} + 36\,n^{2} + 102\,n^{2} x + 106\,n^{2} x^{2} + 48\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2325
358316
4122618460
5165057285156
636132189132456540
7442043873842106071296

All 15 rows

Technical details
Node id
n-c339f708acd8421e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-97e8a3dee23710ef

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   0;
  1:   1;
  2:   3,   2;
  3:   5,   8,    3;
  4:  12,  26,   18,    4;
  5:  16,  50,   57,   28,    5;
  6:  36, 132,  189,  132,   45,    6;
  7:  44, 204,  387,  384,  210,   60,   7;
  8:  96, 496, 1080, 1284,  900,  372,  84,   8;
  9: 112, 688, 1824, 2724, 2505, 1452, 518, 104, 9;
%F Formula
The row polynomials satisfy (9*n^2-23*n+14+(12*n^2-34*n+22)*x+(4*n^2-12*n+8)*x^2)*P(n,x) = -(27*n^3-111*n^2+112*n-8+(54*n^3-227*n^2+243*n-24)*x+(36*n^3-156*n^2+178*n-24)*x^2+(8*n^3-36*n^2+44*n-8)*x^3)*P(n-1,x) + (36*n^2-60*n+8+(102*n^2-178*n+28)*x+(106*n^2-194*n+36)*x^2+(48*n^2-92*n+20)*x^3+(8*n^2-16*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 111*n^2 + 112*n - 8 + (54*n^3 - 227*n^2 + 243*n - 24)*x + (36*n^3 - 156*n^2 + 178*n - 24)*x^2 + (8*n^3 - 36*n^2 + 44*n - 8)*x^3)*P[n - 1, x] + (36*n^2 - 60*n + 8 + (102*n^2 - 178*n + 28)*x + (106*n^2 - 194*n + 36)*x^2 + (48*n^2 - 92*n + 20)*x^3 + (8*n^2 - 16*n + 4)*x^4)*D[P[n - 1, x], x])/(9*n^2 - 23*n + 14 + (12*n^2 - 34*n + 22)*x + (4*n^2 - 12*n + 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(27*n^3 - 111*n^2 + 112*n - 8 + (54*n^3 - 227*n^2 + 243*n - 24)*x + (36*n^3 - 156*n^2 + 178*n - 24)*x^2 + (8*n^3 - 36*n^2 + 44*n - 8)*x^3)*v[i - 1] + (36*n^2 - 60*n + 8 + (102*n^2 - 178*n + 28)*x + (106*n^2 - 194*n + 36)*x^2 + (48*n^2 - 92*n + 20)*x^3 + (8*n^2 - 16*n + 4)*x^4)*deriv(v[i - 1], x))/(9*n^2 - 23*n + 14 + (12*n^2 - 34*n + 22)*x + (4*n^2 - 12*n + 8)*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

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.