Real-rooted atlas

A008277 → θ+1 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\right)\,P_{n}(x) = \left(-194 - 288\,x - 278\,x^{2} + 40\,x^{3} + 129\,n + 161\,n x + 82\,n x^{2} + 52\,n x^{3} - 18\,n^{2} - 21\,n^{2} x - 5\,n^{2} x^{2} - 27\,n^{2} x^{3} + 3\,n^{3} x^{3}\right)\,P_{n-4}(x) + \left(18 - 144\,x - 184\,x^{2} - 46\,x^{3} + 8\,x^{4} - 9\,n + 99\,n x + 21\,n x^{2} + 92\,n x^{3} + 12\,n x^{4} - 9\,n^{2} x + 3\,n^{2} x^{2} - 33\,n^{2} x^{3} - 3\,n^{2} x^{4} + 3\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(-18\,x^{2} - 18\,x^{3} + 9\,n x^{2} + 12\,n x^{4} - 3\,n^{2} x^{4}\right)\,P_{n-4}''(x) + \left(104 + 82\,x + 242\,x^{2} - 88\,x^{3} - 72\,n - 171\,n x - 98\,n x^{2} - 12\,n x^{3} + 18\,n^{2} + 33\,n^{2} x + 10\,n^{2} x^{2} + 21\,n^{2} x^{3} - 3\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-36 + 90\,x + 152\,x^{2} + 2\,x^{3} - 24\,x^{4} + 18\,n - 81\,n x - 15\,n x^{2} - 68\,n x^{3} - 12\,n x^{4} + 9\,n^{2} x - 6\,n^{2} x^{2} + 27\,n^{2} x^{3} + 3\,n^{2} x^{4} - 3\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(18\,x^{2} + 18\,x^{3} - 8\,x^{4} - 9\,n x^{2} - 12\,n x^{4} + 3\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(6 + 142\,x - 64\,x^{2} + 32\,x^{3} - 27\,n + 12\,n x + 16\,n x^{2} - 16\,n x^{3} - 12\,n^{2} x - 5\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(18 + 18\,x - 4\,x^{2} + 28\,x^{3} + 16\,x^{4} - 9\,n - 6\,n x^{2} + 3\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(8\,x^{4}\right)\,P_{n-2}''(x) + \left(20 - 44\,x + 16\,x^{2} + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)
\(P_{3}(x) = 9 + 8\,x + x^{2}\)
\(P_{4}(x) = 22 + 46\,x + 17\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
398118
4224617186
545190165341435
68159510705186712332
71331526490050121533132113237
820433881734632466212944422261179382
929767565086215489619015585530126455181501660
10415124051294805890081200087103114533235036166103113332088
1156121340295515188667658325198478822531465012670241039172056123203081
1273834826618475528933923231076520432445622091226465032478220630051841051168990472
13949544181206491133353227921999825524470443150307235626791212871358517978870874965820211284408489
14119781991222058230836806238608370105000753025844332803390250864218399481361604497567602990256349416395110166663288
1514851197703892070663746726494067583751564960127353154502462253736825578626073130625056382917496400254927436755126132780183650352122
Technical details
Node id
n-d5c94734a249bbad · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7469e0fa44e6298d

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:   3,     1;
   3:   9,     8,      1;
   4:  22,    46,     17,      1;
   5:  45,   190,    165,     34,       1;
   6:  81,   595,   1070,    518,      67,       1;
   7: 133,  1526,   4900,   5012,    1533,     132,      1;
   8: 204,  3388,  17346,  32466,   21294,    4422,    261,     1;
   9: 297,  6756,  50862, 154896,  190155,   85530,  12645,   518,    1;
  10: 415, 12405, 129480, 589008, 1200087, 1031145, 332350, 36166, 1031, 1;
%F Formula
The row polynomials satisfy 8*P(n,x) = -(18*n^2-129*n+194+(21*n^2-161*n+288)*x+(5*n^2-82*n+278)*x^2-(3*n^3-27*n^2+52*n+40)*x^3)*P(n-4,x) - (9*n-18+(9*n^2-99*n+144)*x-(3*n^2+21*n-184)*x^2-(3*n^3-33*n^2+92*n-46)*x^3+(3*n^2-12*n-8)*x^4)*d/dx P(n-4,x) + ((9*n-18)*x^2-18*x^3-(3*n^2-12*n)*x^4)*d^2/dx^2 P(n-4,x) + (18*n^2-72*n+104+(33*n^2-171*n+82)*x+(10*n^2-98*n+242)*x^2-(3*n^3-21*n^2+12*n+88)*x^3)*P(n-3,x) + (18*n-36+(9*n^2-81*n+90)*x-(6*n^2+15*n-152)*x^2-(3*n^3-27*n^2+68*n-2)*x^3+(3*n^2-12*n-24)*x^4)*d/dx P(n-3,x) - ((9*n-18)*x^2-18*x^3-(3*n^2-12*n-8)*x^4)*d^2/dx^2 P(n-3,x) - (27*n-6+(12*n^2-12*n-142)*x+(5*n^2-16*n+64)*x^2+(16*n-32)*x^3)*P(n-2,x) - (9*n-18-18*x-(3*n^2-6*n-4)*x^2-28*x^3-16*x^4)*d/dx P(n-2,x) + 8*x^4*d^2/dx^2 P(n-2,x) + (6*n+20+(16*n-44)*x+16*x^2)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + x, P(3,x) = 9 + 8*x + x^2, P(4,x) = 22 + 46*x + 17*x^2 + x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[3, x_] := 9 + 8*x + x^2;
P[4, x_] := 22 + 46*x + 17*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 129*n + 194 + (21*n^2 - 161*n + 288)*x + (5*n^2 - 82*n + 278)*x^2 - (3*n^3 - 27*n^2 + 52*n + 40)*x^3)*P[n - 4, x] - (9*n - 18 + (9*n^2 - 99*n + 144)*x - (3*n^2 + 21*n - 184)*x^2 - (3*n^3 - 33*n^2 + 92*n - 46)*x^3 + (3*n^2 - 12*n - 8)*x^4)*D[P[n - 4, x], x] + ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n)*x^4)*D[P[n - 4, x], {x, 2}] + (18*n^2 - 72*n + 104 + (33*n^2 - 171*n + 82)*x + (10*n^2 - 98*n + 242)*x^2 - (3*n^3 - 21*n^2 + 12*n + 88)*x^3)*P[n - 3, x] + (18*n - 36 + (9*n^2 - 81*n + 90)*x - (6*n^2 + 15*n - 152)*x^2 - (3*n^3 - 27*n^2 + 68*n - 2)*x^3 + (3*n^2 - 12*n - 24)*x^4)*D[P[n - 3, x], x] - ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n - 8)*x^4)*D[P[n - 3, x], {x, 2}] - (27*n - 6 + (12*n^2 - 12*n - 142)*x + (5*n^2 - 16*n + 64)*x^2 + (16*n - 32)*x^3)*P[n - 2, x] - (9*n - 18 - 18*x - (3*n^2 - 6*n - 4)*x^2 - 28*x^3 - 16*x^4)*D[P[n - 2, x], x] + 8*x^4*D[P[n - 2, x], {x, 2}] + (6*n + 20 + (16*n - 44)*x + 16*x^2)*P[n - 1, x])/(8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; v[3] = 9 + 8*x + x^2; v[4] = 22 + 46*x + 17*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = (-(18*n^2 - 129*n + 194 + (21*n^2 - 161*n + 288)*x + (5*n^2 - 82*n + 278)*x^2 - (3*n^3 - 27*n^2 + 52*n + 40)*x^3)*v[i - 4] - (9*n - 18 + (9*n^2 - 99*n + 144)*x - (3*n^2 + 21*n - 184)*x^2 - (3*n^3 - 33*n^2 + 92*n - 46)*x^3 + (3*n^2 - 12*n - 8)*x^4)*deriv(v[i - 4], x) + ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n)*x^4)*deriv(deriv(v[i - 4], x), x) + (18*n^2 - 72*n + 104 + (33*n^2 - 171*n + 82)*x + (10*n^2 - 98*n + 242)*x^2 - (3*n^3 - 21*n^2 + 12*n + 88)*x^3)*v[i - 3] + (18*n - 36 + (9*n^2 - 81*n + 90)*x - (6*n^2 + 15*n - 152)*x^2 - (3*n^3 - 27*n^2 + 68*n - 2)*x^3 + (3*n^2 - 12*n - 24)*x^4)*deriv(v[i - 3], x) - ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n - 8)*x^4)*deriv(deriv(v[i - 3], x), x) - (27*n - 6 + (12*n^2 - 12*n - 142)*x + (5*n^2 - 16*n + 64)*x^2 + (16*n - 32)*x^3)*v[i - 2] - (9*n - 18 - 18*x - (3*n^2 - 6*n - 4)*x^2 - 28*x^3 - 16*x^4)*deriv(v[i - 2], x) + 8*x^4*deriv(deriv(v[i - 2], x), x) + (6*n + 20 + (16*n - 44)*x + 16*x^2)*v[i - 1])/(8)); 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. [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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.