Real-rooted atlas

A197208 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(5 + 15\,x + 15\,x^{2} + 5\,x^{3} + 6\,n + 18\,n x + 18\,n x^{2} + 6\,n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n}(x) = \left(20 + 100\,x + 180\,x^{2} + 140\,x^{3} + 40\,x^{4} + 9\,n + 39\,n x + 45\,n x^{2} - 21\,n x^{3} - 78\,n x^{4} - 54\,n x^{5} - 12\,n x^{6} + n^{2} + 11\,n^{2} x + 45\,n^{2} x^{2} + 91\,n^{2} x^{3} + 98\,n^{2} x^{4} + 54\,n^{2} x^{5} + 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(27\,x + 132\,x^{2} + 282\,x^{3} + 348\,x^{4} + 267\,x^{5} + 120\,x^{6} + 24\,x^{7} + 3\,n x - 12\,n x^{2} - 102\,n x^{3} - 228\,n x^{4} - 237\,n x^{5} - 120\,n x^{6} - 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 42\,x^{3} + 120\,x^{4} + 180\,x^{5} + 150\,x^{6} + 66\,x^{7} + 12\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 30 + 180\,x + 180\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
130180180390
260780198013204140
310525201218019320966043785
416867205460015456017724070896464184
525215624197820887040175014015679445226484941468
6360327606123604064760122598001845396013550040387144052845480
7495633601677060157449606797637015279264018537750011523996028809990567682335
8660114840416196053416440316285200994732200177255540017983548009689328002153184006123872700

All 15 rows

Technical details
Node id
n-6130243e51f689b8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-bb0a136ce054ce63

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:   30,    180,      180;
   2:   60,    780,     1980,      1320;
   3:  105,   2520,    12180,     19320,       9660;
   4:  168,   6720,    54600,    154560,     177240,       70896;
   5:  252,  15624,   197820,    887040,    1750140,     1567944,      522648;
   6:  360,  32760,   612360,   4064760,   12259800,    18453960,    13550040,      3871440;
   7:  495,  63360,  1677060,  15744960,   67976370,   152792640,   185377500,    115239960,     28809990;
   8:  660, 114840,  4161960,  53416440,  316285200,   994732200,  1772555400,   1798354800,    968932800,    215318400;
   9:  858, 197340,  9523800, 162625320, 1280539260,  5404679280, 13257644400,  19512979200,  16995092400,   8077898400,  1615579680;
  10: 1092, 324324, 20360340, 452251800, 4624139520, 25429884480, 82395593280, 164915150400, 206395469280, 157382665440, 66910203360, 12165491520;
%F Formula
The row polynomials satisfy (n^2+6*n+5+(3*n^2+18*n+15)*x+(3*n^2+18*n+15)*x^2+(n^2+6*n+5)*x^3)*P(n,x) = (n^2+9*n+20+(11*n^2+39*n+100)*x+(45*n^2+45*n+180)*x^2+(91*n^2-21*n+140)*x^3+(98*n^2-78*n+40)*x^4+(54*n^2-54*n)*x^5+(12*n^2-12*n)*x^6)*P(n-1,x) + ((3*n+27)*x-(12*n-132)*x^2-(102*n-282)*x^3-(228*n-348)*x^4-(237*n-267)*x^5-(120*n-120)*x^6-(24*n-24)*x^7)*d/dx P(n-1,x) + (6*x^2+42*x^3+120*x^4+180*x^5+150*x^6+66*x^7+12*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 30 + 180*x + 180*x^2.
%t Mathematica
P[1, x_] := 30 + 180*x + 180*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 9*n + 20 + (11*n^2 + 39*n + 100)*x + (45*n^2 + 45*n + 180)*x^2 + (91*n^2 - 21*n + 140)*x^3 + (98*n^2 - 78*n + 40)*x^4 + (54*n^2 - 54*n)*x^5 + (12*n^2 - 12*n)*x^6)*P[n - 1, x] + ((3*n + 27)*x - (12*n - 132)*x^2 - (102*n - 282)*x^3 - (228*n - 348)*x^4 - (237*n - 267)*x^5 - (120*n - 120)*x^6 - (24*n - 24)*x^7)*D[P[n - 1, x], x] + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 6*n + 5 + (3*n^2 + 18*n + 15)*x + (3*n^2 + 18*n + 15)*x^2 + (n^2 + 6*n + 5)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 30 + 180*x + 180*x^2; for(n = 2, N, my(i = n); v[i] = ((n^2 + 9*n + 20 + (11*n^2 + 39*n + 100)*x + (45*n^2 + 45*n + 180)*x^2 + (91*n^2 - 21*n + 140)*x^3 + (98*n^2 - 78*n + 40)*x^4 + (54*n^2 - 54*n)*x^5 + (12*n^2 - 12*n)*x^6)*v[i - 1] + ((3*n + 27)*x - (12*n - 132)*x^2 - (102*n - 282)*x^3 - (228*n - 348)*x^4 - (237*n - 267)*x^5 - (120*n - 120)*x^6 - (24*n - 24)*x^7)*deriv(v[i - 1], x) + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 6*n + 5 + (3*n^2 + 18*n + 15)*x + (3*n^2 + 18*n + 15)*x^2 + (n^2 + 6*n + 5)*x^3)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023