Real-rooted atlas

A152842 → reverse → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 9\,x + 2\,x^{2} - 13\,n - 13\,n x - 3\,n x^{2} + 4\,n^{2} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(54 + 54\,x + 18\,x^{2} + 2\,x^{3} - 141\,n - 178\,n x - 76\,n x^{2} - 11\,n x^{3} + 88\,n^{2} + 122\,n^{2} x + 57\,n^{2} x^{2} + 9\,n^{2} x^{3} - 16\,n^{3} - 24\,n^{3} x - 12\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(27 + 54\,x + 36\,x^{2} + 10\,x^{3} + x^{4} - 60\,n - 134\,n x - 104\,n x^{2} - 34\,n x^{3} - 4\,n x^{4} + 24\,n^{2} + 56\,n^{2} x + 46\,n^{2} x^{2} + 16\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2426
31514332
4244424496
581180138445448
61083424082286061152
73511242173712284659075120

All 15 rows

Technical details
Node id
n-ba04e3e0da3e4c62 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3ec4d21aec155d88

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:    4,    2;
  3:   15,   14,     3;
  4:   24,   44,    24,     4;
  5:   81,  180,   138,    44,     5;
  6:  108,  342,   408,   228,    60,    6;
  7:  351, 1242,  1737,  1228,   465,   90,    7;
  8:  432, 1944,  3600,  3544,  2000,  648,  112,   8;
  9: 1377, 6696, 13716, 15432, 10430, 4344, 1092, 152, 9;
%F Formula
The row polynomials satisfy (4*n^2-13*n+9+(4*n^2-13*n+9)*x+(n^2-3*n+2)*x^2)*P(n,x) = -(16*n^3-88*n^2+141*n-54+(24*n^3-122*n^2+178*n-54)*x+(12*n^3-57*n^2+76*n-18)*x^2+(2*n^3-9*n^2+11*n-2)*x^3)*P(n-1,x) + (24*n^2-60*n+27+(56*n^2-134*n+54)*x+(46*n^2-104*n+36)*x^2+(16*n^2-34*n+10)*x^3+(2*n^2-4*n+1)*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[(-(16*n^3 - 88*n^2 + 141*n - 54 + (24*n^3 - 122*n^2 + 178*n - 54)*x + (12*n^3 - 57*n^2 + 76*n - 18)*x^2 + (2*n^3 - 9*n^2 + 11*n - 2)*x^3)*P[n - 1, x] + (24*n^2 - 60*n + 27 + (56*n^2 - 134*n + 54)*x + (46*n^2 - 104*n + 36)*x^2 + (16*n^2 - 34*n + 10)*x^3 + (2*n^2 - 4*n + 1)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 13*n + 9 + (4*n^2 - 13*n + 9)*x + (n^2 - 3*n + 2)*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] = (-(16*n^3 - 88*n^2 + 141*n - 54 + (24*n^3 - 122*n^2 + 178*n - 54)*x + (12*n^3 - 57*n^2 + 76*n - 18)*x^2 + (2*n^3 - 9*n^2 + 11*n - 2)*x^3)*v[i - 1] + (24*n^2 - 60*n + 27 + (56*n^2 - 134*n + 54)*x + (46*n^2 - 104*n + 36)*x^2 + (16*n^2 - 34*n + 10)*x^3 + (2*n^2 - 4*n + 1)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 13*n + 9 + (4*n^2 - 13*n + 9)*x + (n^2 - 3*n + 2)*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.