Real-rooted atlas

A086645 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-72 + 468\,x - 162\,x^{2} + 132\,n - 1194\,n x + 621\,n x^{2} - 80\,n^{2} + 1128\,n^{2} x - 936\,n^{2} x^{2} + 16\,n^{3} - 468\,n^{3} x + 693\,n^{3} x^{2} + 72\,n^{4} x - 252\,n^{4} x^{2} + 36\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-288 - 720\,x + 360\,x^{2} + 108\,x^{3} + 528\,n - 24\,n x - 4464\,n x^{2} - 810\,n x^{3} - 320\,n^{2} + 1632\,n^{2} x + 6534\,n^{2} x^{2} + 64\,n^{3} - 1296\,n^{3} x - 2916\,n^{3} x^{2} + 1701\,n^{3} x^{3} + 288\,n^{4} x + 72\,n^{4} x^{2} - 1404\,n^{4} x^{3} + 144\,n^{5} x^{2} + 324\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-72\,x - 204\,x^{2} + 54\,x^{3} + 54\,x^{4} + 96\,n x - 48\,n x^{2} - 1134\,n x^{3} - 378\,n x^{4} - 32\,n^{2} x + 336\,n^{2} x^{2} + 1314\,n^{2} x^{3} + 54\,n^{2} x^{4} - 144\,n^{3} x^{2} - 288\,n^{3} x^{3} + 432\,n^{3} x^{4} - 72\,n^{4} x^{3} - 216\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221214
389681185
43257614584322498
512830721296013824202532009
651215360907201728001012508748389390
720487372855987215482881701000629856357214550513
881923440643193344116121601905120013716864350065813996851566450

All 15 rows

Technical details
Node id
n-9de5cc9733bfaf80 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a33b029f191e5a45

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:      2,      12;
   3:      8,      96,       81;
   4:     32,     576,     1458,       432;
   5:    128,    3072,    12960,     13824,       2025;
   6:    512,   15360,    90720,    172800,     101250,       8748;
   7:   2048,   73728,   559872,   1548288,    1701000,     629856,      35721;
   8:   8192,  344064,  3193344,  11612160,   19051200,   13716864,    3500658,    139968;
   9:  32768, 1572864, 17252352,  77856768,  171072000,  188116992,   96018048,  17915904,   531441;
  10: 131072, 7077888, 89579520, 483065856, 1334361600, 1995383808, 1584297792, 604661760, 86093442, 1968300;
%F Formula
The row polynomials satisfy (16*n^3-80*n^2+132*n-72+(72*n^4-468*n^3+1128*n^2-1194*n+468)*x+(36*n^5-252*n^4+693*n^3-936*n^2+621*n-162)*x^2)*P(n,x) = (64*n^3-320*n^2+528*n-288+(288*n^4-1296*n^3+1632*n^2-24*n-720)*x+(144*n^5+72*n^4-2916*n^3+6534*n^2-4464*n+360)*x^2+(324*n^5-1404*n^4+1701*n^3-810*n+108)*x^3)*P(n-1,x) - ((32*n^2-96*n+72)*x+(144*n^3-336*n^2+48*n+204)*x^2+(72*n^4+288*n^3-1314*n^2+1134*n-54)*x^3+(216*n^4-432*n^3-54*n^2+378*n-54)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((64*n^3 - 320*n^2 + 528*n - 288 + (288*n^4 - 1296*n^3 + 1632*n^2 - 24*n - 720)*x + (144*n^5 + 72*n^4 - 2916*n^3 + 6534*n^2 - 4464*n + 360)*x^2 + (324*n^5 - 1404*n^4 + 1701*n^3 - 810*n + 108)*x^3)*P[n - 1, x] - ((32*n^2 - 96*n + 72)*x + (144*n^3 - 336*n^2 + 48*n + 204)*x^2 + (72*n^4 + 288*n^3 - 1314*n^2 + 1134*n - 54)*x^3 + (216*n^4 - 432*n^3 - 54*n^2 + 378*n - 54)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 80*n^2 + 132*n - 72 + (72*n^4 - 468*n^3 + 1128*n^2 - 1194*n + 468)*x + (36*n^5 - 252*n^4 + 693*n^3 - 936*n^2 + 621*n - 162)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 12*x; for(n = 3, N, my(i = n); v[i] = ((64*n^3 - 320*n^2 + 528*n - 288 + (288*n^4 - 1296*n^3 + 1632*n^2 - 24*n - 720)*x + (144*n^5 + 72*n^4 - 2916*n^3 + 6534*n^2 - 4464*n + 360)*x^2 + (324*n^5 - 1404*n^4 + 1701*n^3 - 810*n + 108)*x^3)*v[i - 1] - ((32*n^2 - 96*n + 72)*x + (144*n^3 - 336*n^2 + 48*n + 204)*x^2 + (72*n^4 + 288*n^3 - 1314*n^2 + 1134*n - 54)*x^3 + (216*n^4 - 432*n^3 - 54*n^2 + 378*n - 54)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 80*n^2 + 132*n - 72 + (72*n^4 - 468*n^3 + 1128*n^2 - 1194*n + 468)*x + (36*n^5 - 252*n^4 + 693*n^3 - 936*n^2 + 621*n - 162)*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023