Real-rooted atlas

A280580 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-36\,n + 6\,n x + 72\,n^{2} - 46\,n^{2} x + 90\,n^{3} x - n^{3} x^{2} - 44\,n^{4} x + 4\,n^{4} x^{2} - 5\,n^{5} x^{2} + 2\,n^{6} x^{2}\right)\,P_{n}(x) = \left(216 + 828\,x + 228\,x^{2} + 6\,x^{3} - 576\,n - 1428\,n x - 350\,n x^{2} - 13\,n x^{3} + 288\,n^{2} + 140\,n^{2} x - 168\,n^{2} x^{2} - 3\,n^{2} x^{3} + 624\,n^{3} x + 496\,n^{3} x^{2} + 24\,n^{3} x^{3} - 176\,n^{4} x - 182\,n^{4} x^{2} - 12\,n^{4} x^{3} - 32\,n^{5} x^{2} - 8\,n^{5} x^{3} + 8\,n^{6} x^{2} + 6\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-432\,x - 156\,x^{2} + 54\,x^{3} + 3\,x^{4} + 576\,n x + 152\,n x^{2} - 54\,n x^{3} - 5\,n x^{4} + 280\,n^{2} x^{2} - 54\,n^{2} x^{3} - 4\,n^{2} x^{4} - 352\,n^{3} x^{2} + 48\,n^{3} x^{3} + 10\,n^{3} x^{4} + 4\,n^{4} x^{3} - n^{4} x^{4} + 16\,n^{5} x^{3} - 4\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
3224935
4512013516276
5145601260448252307
64225209450672011253619893
713211088623707392024750237649174685
842948048378378672672375375720724459641551497
91430205920216216053813764504500144144017836076808113882947
10486287516011814660392071684594590022054032454818039168012393100124854135
11167963695120623551502660486404157010002793510728641542012403200784890190001211126790409
12587861551950432008977017071454403429533250307286179213307974682865139203021826514630002795114410194229290
132080126489974416062686641047049203226293088250302896376641749048100852718561288340241146729800025714923974416992390865021
1474290027041560079096563006187108928018989452625027344811780020301451170081588249600180705224702187185000139289150430560054925196838398666771
15267444011232648003833141130035435260224013050932895002296964189520213165237285011014413696003252694044605511706200052651298702712528006920550740882257614891018243
Technical details
Node id
n-6edcfb410ae9149a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-efa7f5c452f8b8a4

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,      4;
   3:    2,     24,        9;
   4:    5,    120,      135,       16;
   5:   14,    560,     1260,      448,       25;
   6:   42,   2520,     9450,     6720,     1125,       36;
   7:  132,  11088,    62370,    73920,    24750,     2376,      49;
   8:  429,  48048,   378378,   672672,   375375,    72072,    4459,     64;
   9: 1430, 205920,  2162160,  5381376,  4504500,  1441440,  178360,   7680,    81;
  10: 4862, 875160, 11814660, 39207168, 45945900, 22054032, 4548180, 391680, 12393, 100;
%F Formula
The row polynomials satisfy (72*n^2-36*n-(44*n^4-90*n^3+46*n^2-6*n)*x+(2*n^6-5*n^5+4*n^4-n^3)*x^2)*P(n,x) = (288*n^2-576*n+216-(176*n^4-624*n^3-140*n^2+1428*n-828)*x+(8*n^6-32*n^5-182*n^4+496*n^3-168*n^2-350*n+228)*x^2+(6*n^6-8*n^5-12*n^4+24*n^3-3*n^2-13*n+6)*x^3)*P(n-1,x) + ((576*n-432)*x-(352*n^3-280*n^2-152*n+156)*x^2+(16*n^5+4*n^4+48*n^3-54*n^2-54*n+54)*x^3-(4*n^5+n^4-10*n^3+4*n^2+5*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((288*n^2 - 576*n + 216 - (176*n^4 - 624*n^3 - 140*n^2 + 1428*n - 828)*x + (8*n^6 - 32*n^5 - 182*n^4 + 496*n^3 - 168*n^2 - 350*n + 228)*x^2 + (6*n^6 - 8*n^5 - 12*n^4 + 24*n^3 - 3*n^2 - 13*n + 6)*x^3)*P[n - 1, x] + ((576*n - 432)*x - (352*n^3 - 280*n^2 - 152*n + 156)*x^2 + (16*n^5 + 4*n^4 + 48*n^3 - 54*n^2 - 54*n + 54)*x^3 - (4*n^5 + n^4 - 10*n^3 + 4*n^2 + 5*n - 3)*x^4)*D[P[n - 1, x], x])/(72*n^2 - 36*n - (44*n^4 - 90*n^3 + 46*n^2 - 6*n)*x + (2*n^6 - 5*n^5 + 4*n^4 - n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((288*n^2 - 576*n + 216 - (176*n^4 - 624*n^3 - 140*n^2 + 1428*n - 828)*x + (8*n^6 - 32*n^5 - 182*n^4 + 496*n^3 - 168*n^2 - 350*n + 228)*x^2 + (6*n^6 - 8*n^5 - 12*n^4 + 24*n^3 - 3*n^2 - 13*n + 6)*x^3)*v[i - 1] + ((576*n - 432)*x - (352*n^3 - 280*n^2 - 152*n + 156)*x^2 + (16*n^5 + 4*n^4 + 48*n^3 - 54*n^2 - 54*n + 54)*x^3 - (4*n^5 + n^4 - 10*n^3 + 4*n^2 + 5*n - 3)*x^4)*deriv(v[i - 1], x))/(72*n^2 - 36*n - (44*n^4 - 90*n^3 + 46*n^2 - 6*n)*x + (2*n^6 - 5*n^5 + 4*n^4 - n^3)*x^2)); 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.