Real-rooted atlas

A269939 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 6\,n + n^{2}\right)\,P_{n}(x) = \left(8\,x + 288\,x^{2} - 4\,n x - 498\,n x^{2} + 317\,n^{2} x^{2} - 88\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-x - 26\,x^{2} - 81\,x^{3} + 19\,n x^{2} + 36\,n x^{3} - 4\,n^{2} x^{2} + 39\,n^{2} x^{3} - 26\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-7\,x^{2} - 70\,x^{3} - 63\,x^{4} + 5\,n x^{2} + 83\,n x^{3} + 78\,n x^{4} - n^{2} x^{2} - 32\,n^{2} x^{3} - 31\,n^{2} x^{4} + 4\,n^{3} x^{3} + 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-6 - 36\,x + 5\,n + 60\,n x - n^{2} - 28\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(16\,x + 72\,x^{2} - 11\,n x - 84\,n x^{2} + 2\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4\,x\)
\(P_{3}(x) = 4\,x + 27\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2044
3042731
40490240334
504225168026254534
6045047840315003402073868
7041071306882362506237005093551401068
80422141092001424500686070013243230864864030288488
9044509366960757487558918860200855655302702400164189025734612288
10049108118905637180000436395960232786554060540480007443235800344594250019745865968
11041831537614721728727002931528600228769440909102597504018980251290019641872250079222218075582454553696
12043673811707696773773000183916933202011038429401145143359360359210559708062199262125005545555265250197990072280018702912210688
1304735933603745633694911251097861965201632186405850127439959046405627396399625014467330127250021350387771212516697162762280053443570205025649355714917888
140414731211007484814378900900631416125340124828802498401297128421388807727741083062002761014333577500598624205699520076806948706488005344357020502500154954730180250024237471319469824
150429475933441475260439466750352978173802891226404073805123407329741696096297169989220204583805540063750013670144947341382525546659026288400028948600527721875018129703431089250048028060502296875967775802483670528
Technical details
Node id
n-35cc86536da6692a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-be569a6fd03aad81

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, 4;
   3: 0, 4,   27;
   4: 0, 4,   90,     240;
   5: 0, 4,  225,    1680,     2625;
   6: 0, 4,  504,    7840,    31500,     34020;
   7: 0, 4, 1071,   30688,   236250,    623700,     509355;
   8: 0, 4, 2214,  109200,  1424500,   6860700,   13243230,    8648640;
   9: 0, 4, 4509,  366960,  7574875,  58918860,  200855655,  302702400,  164189025;
  10: 0, 4, 9108, 1189056, 37180000, 436395960, 2327865540, 6054048000, 7443235800, 3445942500;
%F Formula
The row polynomials satisfy (n^2-6*n+9)*P(n,x) = -((4*n-8)*x-(9*n^4-88*n^3+317*n^2-498*n+288)*x^2)*P(n-2,x) - (x+(4*n^2-19*n+26)*x^2-(4*n^4-26*n^3+39*n^2+36*n-81)*x^3)*d/dx P(n-2,x) - ((n^2-5*n+7)*x^2-(4*n^3-32*n^2+83*n-70)*x^3-(4*n^3-31*n^2+78*n-63)*x^4)*d^2/dx^2 P(n-2,x) - (n^2-5*n+6-(4*n^3-28*n^2+60*n-36)*x)*P(n-1,x) + ((2*n^2-11*n+16)*x-(4*n^3-32*n^2+84*n-72)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 4*x + 27*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4*x;
P[3, x_] := 4*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n - 8)*x - (9*n^4 - 88*n^3 + 317*n^2 - 498*n + 288)*x^2)*P[n - 2, x] - (x + (4*n^2 - 19*n + 26)*x^2 - (4*n^4 - 26*n^3 + 39*n^2 + 36*n - 81)*x^3)*D[P[n - 2, x], x] - ((n^2 - 5*n + 7)*x^2 - (4*n^3 - 32*n^2 + 83*n - 70)*x^3 - (4*n^3 - 31*n^2 + 78*n - 63)*x^4)*D[P[n - 2, x], {x, 2}] - (n^2 - 5*n + 6 - (4*n^3 - 28*n^2 + 60*n - 36)*x)*P[n - 1, x] + ((2*n^2 - 11*n + 16)*x - (4*n^3 - 32*n^2 + 84*n - 72)*x^2)*D[P[n - 1, x], x])/(n^2 - 6*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; v[3] = 4*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-((4*n - 8)*x - (9*n^4 - 88*n^3 + 317*n^2 - 498*n + 288)*x^2)*v[i - 2] - (x + (4*n^2 - 19*n + 26)*x^2 - (4*n^4 - 26*n^3 + 39*n^2 + 36*n - 81)*x^3)*deriv(v[i - 2], x) - ((n^2 - 5*n + 7)*x^2 - (4*n^3 - 32*n^2 + 83*n - 70)*x^3 - (4*n^3 - 31*n^2 + 78*n - 63)*x^4)*deriv(deriv(v[i - 2], x), x) - (n^2 - 5*n + 6 - (4*n^3 - 28*n^2 + 60*n - 36)*x)*v[i - 1] + ((2*n^2 - 11*n + 16)*x - (4*n^3 - 32*n^2 + 84*n - 72)*x^2)*deriv(v[i - 1], x))/(n^2 - 6*n + 9)); 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.