Real-rooted atlas

A105278 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 27\,x + 41\,x^{2} + 26\,x^{3} + 6\,x^{4} - 2\,n - 13\,n x - 25\,n x^{2} - 19\,n x^{3} - 5\,n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(2 + 16\,x + 42\,x^{2} + 50\,x^{3} + 28\,x^{4} + 6\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
36181337
42410815673361
5120720156014605014361
67205400156002190015030405162701
7504045360163800306600315630170142376331044205
8403204233601834560429240058917604763976210744839435319748177

All 15 rows

Technical details
Node id
n-d4794c9fc15957a0 · 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:       2,        3;
   3:       6,       18,        13;
   4:      24,      108,       156,        73;
   5:     120,      720,      1560,      1460,        501;
   6:     720,     5400,     15600,     21900,      15030,       4051;
   7:    5040,    45360,    163800,    306600,     315630,     170142,      37633;
   8:   40320,   423360,   1834560,   4292400,    5891760,    4763976,    2107448,     394353;
   9:  362880,  4354560,  22014720,  61810560,  106051680,  114335424,   75868128,   28393416,   4596553;
  10: 3628800, 48988800, 283046400, 927158400, 1908930240, 2572547040, 2276043840, 1277703720, 413689770, 58941091;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n-6-(n^2-13*n+27)*x-(3*n^2-25*n+41)*x^2-(3*n^2-19*n+26)*x^3-(n^2-5*n+6)*x^4)*P(n-1,x) + (2-(2*n-16)*x-(8*n-42)*x^2-(12*n-50)*x^3-(8*n-28)*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[(-(2*n - 6 - (n^2 - 13*n + 27)*x - (3*n^2 - 25*n + 41)*x^2 - (3*n^2 - 19*n + 26)*x^3 - (n^2 - 5*n + 6)*x^4)*P[n - 1, x] + (2 - (2*n - 16)*x - (8*n - 42)*x^2 - (12*n - 50)*x^3 - (8*n - 28)*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] = (-(2*n - 6 - (n^2 - 13*n + 27)*x - (3*n^2 - 25*n + 41)*x^2 - (3*n^2 - 19*n + 26)*x^3 - (n^2 - 5*n + 6)*x^4)*v[i - 1] + (2 - (2*n - 16)*x - (8*n - 42)*x^2 - (12*n - 50)*x^3 - (8*n - 28)*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.