Real-rooted atlas

A028421 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 12\,x - x^{2} + 4\,n - 16\,n x + 2\,n x^{2} + 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 4\,x^{2} - 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(8 - x - 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
2912223
31654446120
42516031520024724
5363751360202510961205012
64975643751176014616705672039332
76413721159249000108304118188522725040345832
88123042675416329656122510765441063116438336403203371976
9100364555680463050227782867331251157888010554300410630436288036135792
101215500106920116160077308773247398085423250134552000114782184425145603628800422379792
111447986192500264627022875072129240342480223260114989325016841292161358261784482175360399168005349561984
121691123232875755770006070650344330457622045213197418525400164301709002262423820817384035968594577152047900160072996193152
13196153795372641102201114744730013502418938642334272387562144971201002762962489425939003250200495362393704535047921103616062270208001067779243008
142252058084584520638800332719387373122750029841340815171613041600706056823472203988120336039930401420004975828166400352941118041611338625894408717829120016670798231040
Technical details
Node id
n-815236776f18362c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-12b23884bf72db83

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,    1;
  2:   9,   12,     2;
  3:  16,   54,    44,      6;
  4:  25,  160,   315,    200,      24;
  5:  36,  375,  1360,   2025,    1096,     120;
  6:  49,  756,  4375,  11760,   14616,    7056,      720;
  7:  64, 1372, 11592,  49000,  108304,  118188,    52272,     5040;
  8:  81, 2304, 26754, 163296,  561225, 1076544,  1063116,   438336,   40320;
  9: 100, 3645, 55680, 463050, 2277828, 6733125, 11578880, 10554300, 4106304, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+(4*n^2-16*n+12)*x-(n^2-2*n+1)*x^2)*P(n-2,x) - (4*x+(4*n-4)*x^2)*d/dx P(n-2,x) - (4*n-8-(2*n-1)*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(4*n + 1 + (4*n^2 - 16*n + 12)*x - (n^2 - 2*n + 1)*x^2)*P[n - 2, x] - (4*x + (4*n - 4)*x^2)*D[P[n - 2, x], x] - (4*n - 8 - (2*n - 1)*x)*P[n - 1, x] + 4*x*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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (4*n + 1 + (4*n^2 - 16*n + 12)*x - (n^2 - 2*n + 1)*x^2)*v[i - 2] - (4*x + (4*n - 4)*x^2)*deriv(v[i - 2], x) - (4*n - 8 - (2*n - 1)*x)*v[i - 1] + 4*x*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

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.