Real-rooted atlas

A071951 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 15\,x + 38\,x^{2} + 45\,x^{3} + 26\,x^{4} + 6\,x^{5} - 4\,n x - 17\,n x^{2} - 27\,n x^{3} - 19\,n x^{4} - 5\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(4\,x + 22\,x^{2} + 48\,x^{3} + 52\,x^{4} + 28\,x^{5} + 6\,x^{6} - 2\,n x^{2} - 8\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
34161333
487614881313
516384134816486694065
6322096118882450021630695567101
76412032107024334720484568333408885051360321
812870848100793645425449624720106461205972936134620933211441

All 15 rows

Technical details
Node id
n-7abf9f89c09ae763 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-af2a7605ed60a01d

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 -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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   2,       3;
   3:   4,      16,        13;
   4:   8,      76,       148,        81;
   5:  16,     384,      1348,      1648,        669;
   6:  32,    2096,     11888,     24500,      21630,       6955;
   7:  64,   12032,    107024,    334720,     484568,     333408,      88505;
   8: 128,   70848,   1007936,   4542544,    9624720,   10646120,    5972936,    1346209;
   9: 256,  421888,  10042432,  63216896,  184399984,  291980928,  259945144,  122766624,   23998521;
  10: 512, 2523904, 105749248, 912867136, 3535665056, 7533165328, 9474847488, 7039311928, 2865562458, 493956467;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2-(4*n-15)*x+(n^2-17*n+38)*x^2+(3*n^2-27*n+45)*x^3+(3*n^2-19*n+26)*x^4+(n^2-5*n+6)*x^5)*P(n-1,x) + (4*x-(2*n-22)*x^2-(8*n-48)*x^3-(12*n-52)*x^4-(8*n-28)*x^5-(2*n-6)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2 - (4*n - 15)*x + (n^2 - 17*n + 38)*x^2 + (3*n^2 - 27*n + 45)*x^3 + (3*n^2 - 19*n + 26)*x^4 + (n^2 - 5*n + 6)*x^5)*P[n - 1, x] + (4*x - (2*n - 22)*x^2 - (8*n - 48)*x^3 - (12*n - 52)*x^4 - (8*n - 28)*x^5 - (2*n - 6)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2 - (4*n - 15)*x + (n^2 - 17*n + 38)*x^2 + (3*n^2 - 27*n + 45)*x^3 + (3*n^2 - 19*n + 26)*x^4 + (n^2 - 5*n + 6)*x^5)*v[i - 1] + (4*x - (2*n - 22)*x^2 - (8*n - 48)*x^3 - (12*n - 52)*x^4 - (8*n - 28)*x^5 - (2*n - 6)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 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, v[i])))};
T(10)

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.