Real-rooted atlas

A122542 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 12\,x + 81\,x^{2}\right)\,P_{n-4}(x) + \left(12\,x^{2} + 36\,x^{3}\right)\,P_{n-4}'(x) + \left(-4 - 36\,x + 162\,x^{2}\right)\,P_{n-3}(x) + \left(36\,x^{2} + 72\,x^{3}\right)\,P_{n-3}'(x) + \left(-2 - 12\,x + 81\,x^{2}\right)\,P_{n-2}(x) + \left(12\,x^{2} + 36\,x^{3}\right)\,P_{n-2}'(x) + \left(4 + 12\,x\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 12\,x\)
\(P_{3}(x) = 3 + 48\,x + 81\,x^{2}\)
\(P_{4}(x) = 7 + 156\,x + 567\,x^{2} + 432\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211213
334881132
471565674321162
5174802592432020259434
6411404100442592026325874872482
79939843539712355219642513996835721535146
823911052117207515376111982512509646786991399683833330

All 15 rows

Technical details
Node id
n-8843ddf740f215f1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6d17f57f0c924e63

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:    1,    12;
   3:    3,    48,      81;
   4:    7,   156,     567,     432;
   5:   17,   480,    2592,    4320,     2025;
   6:   41,  1404,   10044,   25920,    26325,     8748;
   7:   99,  3984,   35397,  123552,   196425,   139968,    35721;
   8:  239, 11052,  117207,  515376,  1119825,  1250964,   678699,   139968;
   9:  577, 30144,  371304, 1968192,  5431050,  8328096,  7072758,  3079296,   531441;
  10: 1393, 81132, 1137888, 7057152, 23623650, 46206936, 53938710, 36671616, 13286025, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(1+12*x-81*x^2)*P(n-4,x) + (12*x^2+36*x^3)*d/dx P(n-4,x) - (4+36*x-162*x^2)*P(n-3,x) + (36*x^2+72*x^3)*d/dx P(n-3,x) - (2+12*x-81*x^2)*P(n-2,x) + (12*x^2+36*x^3)*d/dx P(n-2,x) + (4+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 12*x, P(3,x) = 3 + 48*x + 81*x^2, P(4,x) = 7 + 156*x + 567*x^2 + 432*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 12*x;
P[3, x_] := 3 + 48*x + 81*x^2;
P[4, x_] := 7 + 156*x + 567*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 12*x - 81*x^2)*P[n - 4, x] + (12*x^2 + 36*x^3)*D[P[n - 4, x], x] - (4 + 36*x - 162*x^2)*P[n - 3, x] + (36*x^2 + 72*x^3)*D[P[n - 3, x], x] - (2 + 12*x - 81*x^2)*P[n - 2, x] + (12*x^2 + 36*x^3)*D[P[n - 2, x], x] + (4 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; v[3] = 3 + 48*x + 81*x^2; v[4] = 7 + 156*x + 567*x^2 + 432*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 12*x - 81*x^2)*v[i - 4] + (12*x^2 + 36*x^3)*deriv(v[i - 4], x) - (4 + 36*x - 162*x^2)*v[i - 3] + (36*x^2 + 72*x^3)*deriv(v[i - 3], x) - (2 + 12*x - 81*x^2)*v[i - 2] + (12*x^2 + 36*x^3)*deriv(v[i - 2], x) + (4 + 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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.