Real-rooted atlas

A033282 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 6\,x + 8\,x^{2} + 2\,n - 3\,n x - 16\,n x^{2} - 6\,n x^{3} - 30\,n^{2} x^{2} - 9\,n^{2} x^{3} + 4\,n^{2} x^{4} - 10\,n^{3} x^{2} - 3\,n^{3} x^{3} + 10\,n^{3} x^{4} + 8\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(12 + 18\,x - 48\,x^{2} - 112\,x^{3} - 48\,x^{4} + 16\,x^{5} + 6\,n + 37\,n x - 108\,n x^{2} - 322\,n x^{3} - 120\,n x^{4} + 72\,n x^{5} + 14\,n^{2} x - 102\,n^{2} x^{2} - 287\,n^{2} x^{3} - 84\,n^{2} x^{4} + 128\,n^{2} x^{5} - 30\,n^{3} x^{2} - 93\,n^{3} x^{3} + 6\,n^{3} x^{4} + 112\,n^{3} x^{5} - 8\,n^{4} x^{3} + 24\,n^{4} x^{4} + 48\,n^{4} x^{5} + 6\,n^{5} x^{4} + 8\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1 - 19\,x^{2} - 48\,x^{3} - 38\,x^{4} + 8\,x^{6} - 28\,n x^{2} - 96\,n x^{3} - 76\,n x^{4} + 24\,n x^{5} + 32\,n x^{6} - 14\,n^{2} x^{2} - 48\,n^{2} x^{3} - 22\,n^{2} x^{4} + 60\,n^{2} x^{5} + 48\,n^{2} x^{6} + 16\,n^{3} x^{4} + 48\,n^{3} x^{5} + 32\,n^{3} x^{6} + 4\,n^{4} x^{4} + 12\,n^{4} x^{5} + 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
211120180311
345744226817924849
4197436820160322561680073781
5903251201555203763203960001520641105927
642791432081112940361152057420004447872134534416407163
72079381340076091403099712066066000772611844708704011714560241569237

All 15 rows

Technical details
Node id
n-deeca0780a643688 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d48d54a9c58ccb5a

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:      3,       16;
  2:     11,      120,       180;
  3:     45,      744,      2268,       1792;
  4:    197,     4368,     20160,      32256,      16800;
  5:    903,    25120,    155520,     376320,     396000,      152064;
  6:   4279,   143208,   1112940,    3611520,    5742000,     4447872,     1345344;
  7:  20793,   813400,   7609140,   30997120,   66066000,    77261184,    47087040,    11714560;
  8: 103049,  4613664,  50472576,  247543296,  662261600,  1037836800,   954297344,   477954048,  100818432;
  9: 518859, 26163504, 327727296, 1880126976, 6056013600, 11910592512, 14670609408, 11079843840, 4701805056, 859955200;
%F Formula
The row polynomials satisfy (2*n+4-(3*n+6)*x-(10*n^3+30*n^2+16*n-8)*x^2-(3*n^3+9*n^2+6*n)*x^3+(2*n^5+8*n^4+10*n^3+4*n^2)*x^4)*P(n,x) = (6*n+12+(14*n^2+37*n+18)*x-(30*n^3+102*n^2+108*n+48)*x^2-(8*n^4+93*n^3+287*n^2+322*n+112)*x^3+(6*n^5+24*n^4+6*n^3-84*n^2-120*n-48)*x^4+(8*n^5+48*n^4+112*n^3+128*n^2+72*n+16)*x^5)*P(n-1,x) + (1-(14*n^2+28*n+19)*x^2-(48*n^2+96*n+48)*x^3+(4*n^4+16*n^3-22*n^2-76*n-38)*x^4+(12*n^4+48*n^3+60*n^2+24*n)*x^5+(8*n^4+32*n^3+48*n^2+32*n+8)*x^6)*d/dx 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[((6*n + 12 + (14*n^2 + 37*n + 18)*x - (30*n^3 + 102*n^2 + 108*n + 48)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (6*n^5 + 24*n^4 + 6*n^3 - 84*n^2 - 120*n - 48)*x^4 + (8*n^5 + 48*n^4 + 112*n^3 + 128*n^2 + 72*n + 16)*x^5)*P[n - 1, x] + (1 - (14*n^2 + 28*n + 19)*x^2 - (48*n^2 + 96*n + 48)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (12*n^4 + 48*n^3 + 60*n^2 + 24*n)*x^5 + (8*n^4 + 32*n^3 + 48*n^2 + 32*n + 8)*x^6)*D[P[n - 1, x], x])/(2*n + 4 - (3*n + 6)*x - (10*n^3 + 30*n^2 + 16*n - 8)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*n^2)*x^4)]];
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] = ((6*n + 12 + (14*n^2 + 37*n + 18)*x - (30*n^3 + 102*n^2 + 108*n + 48)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (6*n^5 + 24*n^4 + 6*n^3 - 84*n^2 - 120*n - 48)*x^4 + (8*n^5 + 48*n^4 + 112*n^3 + 128*n^2 + 72*n + 16)*x^5)*v[i - 1] + (1 - (14*n^2 + 28*n + 19)*x^2 - (48*n^2 + 96*n + 48)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (12*n^4 + 48*n^3 + 60*n^2 + 24*n)*x^5 + (8*n^4 + 32*n^3 + 48*n^2 + 32*n + 8)*x^6)*deriv(v[i - 1], x))/(2*n + 4 - (3*n + 6)*x - (10*n^3 + 30*n^2 + 16*n - 8)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*n^2)*x^4)); 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

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