Real-rooted atlas

A257609 → D (shift +2) → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(-4 - 22\,x - 50\,x^{2} - 58\,x^{3} - 34\,x^{4} - 8\,x^{5} - 2\,n - 13\,n x - 35\,n x^{2} - 47\,n x^{3} - 31\,n x^{4} - 8\,n x^{5} - n^{2} x - 5\,n^{2} x^{2} - 9\,n^{2} x^{3} - 7\,n^{2} x^{4} - 2\,n^{2} x^{5}\right)\,P_{n-2}(x) + \left(-2\,x - 12\,x^{2} - 28\,x^{3} - 32\,x^{4} - 18\,x^{5} - 4\,x^{6} - n x - 8\,n x^{2} - 22\,n x^{3} - 28\,n x^{4} - 17\,n x^{5} - 4\,n x^{6} - n^{2} x^{2} - 4\,n^{2} x^{3} - 6\,n^{2} x^{4} - 4\,n^{2} x^{5} - n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(4 + 18\,x + 30\,x^{2} + 22\,x^{3} + 6\,x^{4} + 3\,n + 15\,n x + 27\,n x^{2} + 21\,n x^{3} + 6\,n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5} + n x + 4\,n x^{2} + 6\,n x^{3} + 4\,n x^{4} + n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 11 + 44\,x + 36\,x^{2}\)
\(P_{2}(x) = 26 + 210\,x + 420\,x^{2} + 240\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
111443691
226210420240896
3578323060408018009829
4120298217976420004200015120120198
524710068934923407045796004636801411201628911
6502327304505402404080618912082555205503680145152024287692
71013103784206370015500880566622001120694401227744007015680016329600395661817
820363235989125952939444004685076001288813680206303328019160064009580032001995840006997344146

All 15 rows

Technical details
Node id
n-3279e81c0b2510f5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b9c53891d5ed77f5

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:   11,      44,        36;
   2:   26,     210,       420,        240;
   3:   57,     832,      3060,       4080,        1800;
   4:  120,    2982,     17976,      42000,       42000,        15120;
   5:  247,   10068,     93492,     340704,      579600,       463680,       141120;
   6:  502,   32730,    450540,    2404080,     6189120,      8255520,      5503680,      1451520;
   7: 1013,  103784,   2063700,   15500880,    56662200,    112069440,    122774400,     70156800,     16329600;
   8: 2036,  323598,   9125952,   93944400,   468507600,   1288813680,   2063033280,   1916006400,    958003200,    199584000;
   9: 4083,  997372,  39355668,  544703808,  3606451200,  13254278400,  29073401280,  39017341440,  31434480000,  13970880000,   2634508800;
  10: 8178, 3049410, 166644660, 3057054000, 26348562240, 125855009280, 363312069120, 660686906880, 762498696960, 542269728000, 216907891200, 37362124800;
%F Formula
The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = -(2*n+4+(n^2+13*n+22)*x+(5*n^2+35*n+50)*x^2+(9*n^2+47*n+58)*x^3+(7*n^2+31*n+34)*x^4+(2*n^2+8*n+8)*x^5)*P(n-2,x) - ((n+2)*x+(n^2+8*n+12)*x^2+(4*n^2+22*n+28)*x^3+(6*n^2+28*n+32)*x^4+(4*n^2+17*n+18)*x^5+(n^2+4*n+4)*x^6)*d/dx P(n-2,x) + (3*n+4+(n^2+15*n+18)*x+(3*n^2+27*n+30)*x^2+(3*n^2+21*n+22)*x^3+(n^2+6*n+6)*x^4)*P(n-1,x) + ((n+1)*x+(4*n+4)*x^2+(6*n+6)*x^3+(4*n+4)*x^4+(n+1)*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 11 + 44*x + 36*x^2, P(2,x) = 26 + 210*x + 420*x^2 + 240*x^3.
%t Mathematica
P[1, x_] := 11 + 44*x + 36*x^2;
P[2, x_] := 26 + 210*x + 420*x^2 + 240*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (n^2 + 13*n + 22)*x + (5*n^2 + 35*n + 50)*x^2 + (9*n^2 + 47*n + 58)*x^3 + (7*n^2 + 31*n + 34)*x^4 + (2*n^2 + 8*n + 8)*x^5)*P[n - 2, x] - ((n + 2)*x + (n^2 + 8*n + 12)*x^2 + (4*n^2 + 22*n + 28)*x^3 + (6*n^2 + 28*n + 32)*x^4 + (4*n^2 + 17*n + 18)*x^5 + (n^2 + 4*n + 4)*x^6)*D[P[n - 2, x], x] + (3*n + 4 + (n^2 + 15*n + 18)*x + (3*n^2 + 27*n + 30)*x^2 + (3*n^2 + 21*n + 22)*x^3 + (n^2 + 6*n + 6)*x^4)*P[n - 1, x] + ((n + 1)*x + (4*n + 4)*x^2 + (6*n + 6)*x^3 + (4*n + 4)*x^4 + (n + 1)*x^5)*D[P[n - 1, x], x])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 11 + 44*x + 36*x^2; v[2] = 26 + 210*x + 420*x^2 + 240*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 4 + (n^2 + 13*n + 22)*x + (5*n^2 + 35*n + 50)*x^2 + (9*n^2 + 47*n + 58)*x^3 + (7*n^2 + 31*n + 34)*x^4 + (2*n^2 + 8*n + 8)*x^5)*v[i - 2] - ((n + 2)*x + (n^2 + 8*n + 12)*x^2 + (4*n^2 + 22*n + 28)*x^3 + (6*n^2 + 28*n + 32)*x^4 + (4*n^2 + 17*n + 18)*x^5 + (n^2 + 4*n + 4)*x^6)*deriv(v[i - 2], x) + (3*n + 4 + (n^2 + 15*n + 18)*x + (3*n^2 + 27*n + 30)*x^2 + (3*n^2 + 21*n + 22)*x^3 + (n^2 + 6*n + 6)*x^4)*v[i - 1] + ((n + 1)*x + (4*n + 4)*x^2 + (6*n + 6)*x^3 + (4*n + 4)*x^4 + (n + 1)*x^5)*deriv(v[i - 1], x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.