Real-rooted atlas

A133087 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x + x^{2} - n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-5 - 4\,x - 2\,x^{2} - 3\,x^{4} - n - 4\,n x + n x^{2} - 4\,n x^{3} + 3\,n x^{4} + 3\,n^{2} + 4\,n^{2} x + n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2 - 6\,x - 10\,x^{2} - 11\,x^{3} - 8\,x^{4} - 3\,x^{5} - 2\,n - 8\,n x - 13\,n x^{2} - 13\,n x^{3} - 6\,n x^{4} - 2\,n^{2} x - 5\,n^{2} x^{2} - 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
23418
351411232
4516188148
57346252192176
6736758045121256
7962177270235114272896
8964196336350224841611280
91198380840116210365882003524352
10111004059601470151210504801352016144
111314269520103810495644942820118531043220480
121314472622004455633664684752247588019824128672
1315194114640929845168302098819272129696270209044451294208
14151961183436811011200202702727456210211201250051456273281131072
Technical details
Node id
n-cedca61b3a4dd93d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-435ec46f3c2a961a

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  3,  2;
  2:  3,  4,   1;
  3:  5, 14,  11,   2;
  4:  5, 16,  18,   8,    1;
  5:  7, 34,  62,  52,   19,    2;
  6:  7, 36,  75,  80,   45,   12,   1;
  7:  9, 62, 177, 270,  235,  114,  27,   2;
  8:  9, 64, 196, 336,  350,  224,  84,  16,  1;
  9: 11, 98, 380, 840, 1162, 1036, 588, 200, 35, 2;
%F Formula
The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = (3*n^2-n-5+(2*n^3+4*n^2-4*n-4)*x+(3*n^3+n^2+n-2)*x^2+(6*n^2-4*n)*x^3+(3*n-3)*x^4)*P(n-1,x) - (2*n+2+(2*n^2+8*n+6)*x+(5*n^2+13*n+10)*x^2+(3*n^2+13*n+11)*x^3+(6*n+8)*x^4+3*x^5)*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[((3*n^2 - n - 5 + (2*n^3 + 4*n^2 - 4*n - 4)*x + (3*n^3 + n^2 + n - 2)*x^2 + (6*n^2 - 4*n)*x^3 + (3*n - 3)*x^4)*P[n - 1, x] - (2*n + 2 + (2*n^2 + 8*n + 6)*x + (5*n^2 + 13*n + 10)*x^2 + (3*n^2 + 13*n + 11)*x^3 + (6*n + 8)*x^4 + 3*x^5)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*x + 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] = ((3*n^2 - n - 5 + (2*n^3 + 4*n^2 - 4*n - 4)*x + (3*n^3 + n^2 + n - 2)*x^2 + (6*n^2 - 4*n)*x^3 + (3*n - 3)*x^4)*v[i - 1] - (2*n + 2 + (2*n^2 + 8*n + 6)*x + (5*n^2 + 13*n + 10)*x^2 + (3*n^2 + 13*n + 11)*x^3 + (6*n + 8)*x^4 + 3*x^5)*deriv(v[i - 1], x))/(n^2 - n - 1 + (2*n - 2)*x + 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.