Real-rooted atlas

A271703 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11\,x + 27\,x^{2} + 22\,x^{3} + 6\,x^{4} - 7\,n x - 19\,n x^{2} - 17\,n x^{3} - 5\,n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 30\,x^{2} + 42\,x^{3} + 26\,x^{4} + 6\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30235
406181337
502410815673361
60120720156014605014361
707205400156002190015030405162701
80504045360163800306600315630170142376331044205

All 15 rows

Technical details
Node id
n-e7c705bd49501967 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-01d09cfb9e6b0947

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: 0,      1;
   3: 0,      2,       3;
   4: 0,      6,      18,       13;
   5: 0,     24,     108,      156,       73;
   6: 0,    120,     720,     1560,     1460,       501;
   7: 0,    720,    5400,    15600,    21900,     15030,      4051;
   8: 0,   5040,   45360,   163800,   306600,    315630,    170142,    37633;
   9: 0,  40320,  423360,  1834560,  4292400,   5891760,   4763976,  2107448,   394353;
  10: 0, 362880, 4354560, 22014720, 61810560, 106051680, 114335424, 75868128, 28393416, 4596553;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = ((n^2-7*n+11)*x+(3*n^2-19*n+27)*x^2+(3*n^2-17*n+22)*x^3+(n^2-5*n+6)*x^4)*P(n-1,x) - ((2*n-8)*x+(8*n-30)*x^2+(12*n-42)*x^3+(8*n-26)*x^4+(2*n-6)*x^5)*d/dx P(n-1,x) + (x+5*x^2+10*x^3+10*x^4+5*x^5+x^6)*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[(((n^2 - 7*n + 11)*x + (3*n^2 - 19*n + 27)*x^2 + (3*n^2 - 17*n + 22)*x^3 + (n^2 - 5*n + 6)*x^4)*P[n - 1, x] - ((2*n - 8)*x + (8*n - 30)*x^2 + (12*n - 42)*x^3 + (8*n - 26)*x^4 + (2*n - 6)*x^5)*D[P[n - 1, x], x] + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
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] = (((n^2 - 7*n + 11)*x + (3*n^2 - 19*n + 27)*x^2 + (3*n^2 - 17*n + 22)*x^3 + (n^2 - 5*n + 6)*x^4)*v[i - 1] - ((2*n - 8)*x + (8*n - 30)*x^2 + (12*n - 42)*x^3 + (8*n - 26)*x^4 + (2*n - 6)*x^5)*deriv(v[i - 1], x) + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); 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.