Real-rooted atlas

A028338 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 + 24\,x + 81\,x^{2} + 8\,n - 24\,n x - 4\,n^{2}\right)\,P_{n-2}(x) + \left(-24\,x^{2} + 36\,x^{3} + 24\,n x^{2}\right)\,P_{n-2}'(x) + \left(-2 + 12\,x + 4\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121214
287281161
3485289724321980
43844800113408640202527189
5384052608145800146880607508748418626
646080677376210470425401601417500367416357217188957
764512010036224340381444678732831752000112674242000376139968136666584

All 15 rows

Technical details
Node id
n-d5996ebbf963d51d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9b2962a65732d26e

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:         2,         12;
  2:         8,         72,          81;
  3:        48,        528,         972,         432;
  4:       384,       4800,       11340,        8640,        2025;
  5:      3840,      52608,      145800,      146880,       60750,       8748;
  6:     46080,     677376,     2104704,     2540160,     1417500,     367416,      35721;
  7:    645120,   10036224,    34038144,    46787328,    31752000,   11267424,    2000376,    139968;
  8:  10321920,  168321024,   612354816,   930134016,   727347600,  317447424,   78014664,  10077696,   531441;
  9: 185794560, 3153641472, 12158553600, 20008304640, 17452260000, 8856195264, 2700507600, 487088640, 47829690, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(24*n-24)*x-81*x^2)*P(n-2,x) + ((24*n-24)*x^2+36*x^3)*d/dx P(n-2,x) + (4*n-2+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (24*n - 24)*x - 81*x^2)*P[n - 2, x] + ((24*n - 24)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (4*n - 2 + 12*x)*P[n - 1, x] - 12*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] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (24*n - 24)*x - 81*x^2)*v[i - 2] + ((24*n - 24)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (4*n - 2 + 12*x)*v[i - 1] - 12*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.