Real-rooted atlas

A156992 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 6\,x^{2} + 2\,x^{3} + 2\,n + 12\,n x + 16\,n x^{2} + 6\,n x^{3} + 6\,n^{2} x + 12\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
224192216432
319223045184307210752
419203072010368012288048000307200
5230404608002073600368640028800008294409953280
63225607741440435456001032192001209600006967296015805440361267200
75160960144506880975421440289013760045158400003901685760177020928033030144014533263360

All 15 rows

Technical details
Node id
n-e642c8ebc6bb8d51 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:          4,          16;
  2:         24,         192,          216;
  3:        192,        2304,         5184,          3072;
  4:       1920,       30720,       103680,        122880,         48000;
  5:      23040,      460800,      2073600,       3686400,       2880000,        829440;
  6:     322560,     7741440,     43545600,     103219200,     120960000,      69672960,      15805440;
  7:    5160960,   144506880,    975421440,    2890137600,    4515840000,    3901685760,    1770209280,     330301440;
  8:   92897280,  2972712960,  23410114560,   83235962880,  162570240000,  187280916480,  127455068160,   47563407360,    7524679680;
  9: 1857945600, 66886041600, 601974374400, 2497078886400, 5852528640000, 8427641241600, 7647304089600, 4280706662400, 1354442342400, 185794560000;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (2*n+2+(6*n^2+12*n+6)*x+(2*n^3+12*n^2+16*n+6)*x^2+(2*n^3+6*n^2+6*n+2)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (6*n^2 + 12*n + 6)*x + (2*n^3 + 12*n^2 + 16*n + 6)*x^2 + (2*n^3 + 6*n^2 + 6*n + 2)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 2 + (6*n^2 + 12*n + 6)*x + (2*n^3 + 12*n^2 + 16*n + 6)*x^2 + (2*n^3 + 6*n^2 + 6*n + 2)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*x^2)); 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

Same fingerprint ?

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.