Real-rooted atlas

A130850 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 7\,x^{2} + 4\,n x - 2\,n x^{2} - n^{2} - 4\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-x - 8\,x^{2} - 13\,x^{3} + 2\,n x - 14\,n x^{3} + 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 4\,x^{3} - 3\,x^{4} - 4\,n x^{3} - 4\,n x^{4}\right)\,P_{n-2}''(x) + \left(1 + 2\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2212923
36486316133
42424045024025979
5120144035102880775368761
67201008030240336001505022684992007
7504080640287280403200255150695526223641107149
8403207257602993760508032041706001678320296450163208115001931
936288072576003402000067737600684810003674160010026870119424041391100225863281
1036288007983360041912640095800320011517660007753838402901544805596800046171621023001213738583903
11399168009580032005568393600143700480002003740200016262104320778307376021187584002971894861730520024768714467452442597
12479001600124540416007939451520022832409600036237801600034466560128020164015872071587676160145965259441519518000633132505896801691317103053603
136227020800174356582400120959879040038358448128006828966144000745374389760051645065472002273093222400616601545560961800840007544406870227223360138427919627666891661865
14871782912002615348736000196151155200006799906713600013421824416000016542080755200013262441512320069875476070400238377945405605048319276000612067135680365955408008027841383211068225621991235077271
Technical details
Node id
n-5ce84951924c70ee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-61423eae1de6db11

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,       4;
  2:      2,      12,        9;
  3:      6,      48,       63,       16;
  4:     24,     240,      450,      240,       25;
  5:    120,    1440,     3510,     2880,      775,       36;
  6:    720,   10080,    30240,    33600,    15050,     2268,       49;
  7:   5040,   80640,   287280,   403200,   255150,    69552,     6223,      64;
  8:  40320,  725760,  2993760,  5080320,  4170600,  1678320,   296450,   16320,    81;
  9: 362880, 7257600, 34020000, 67737600, 68481000, 36741600, 10026870, 1194240, 41391, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-1+(4*n^2-4*n)*x-(9*n^2-2*n-7)*x^2)*P(n-2,x) + ((2*n-1)*x+(4*n^2-8)*x^2+(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) - (x^2+(4*n+4)*x^3+(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+1+4*n*x)*P(n-1,x) - (2*x+(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1 + (4*n^2 - 4*n)*x - (9*n^2 - 2*n - 7)*x^2)*P[n - 2, x] + ((2*n - 1)*x + (4*n^2 - 8)*x^2 + (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] - (x^2 + (4*n + 4)*x^3 + (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + 1 + 4*n*x)*P[n - 1, x] - (2*x + (4*n + 4)*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1 + (4*n^2 - 4*n)*x - (9*n^2 - 2*n - 7)*x^2)*v[i - 2] + ((2*n - 1)*x + (4*n^2 - 8)*x^2 + (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) - (x^2 + (4*n + 4)*x^3 + (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + 1 + 4*n*x)*v[i - 1] - (2*x + (4*n + 4)*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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.