Real-rooted atlas

A063007 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 - 9\,x + 39\,x^{2} + 54\,x^{3} - 16\,x^{4} + 4\,n + 9\,n x - 99\,n x^{2} - 126\,n x^{3} + 64\,n x^{4} + 80\,n^{2} x^{2} + 96\,n^{2} x^{3} - 100\,n^{2} x^{4} - 20\,n^{3} x^{2} - 24\,n^{3} x^{3} + 76\,n^{3} x^{4} - 28\,n^{4} x^{4} + 4\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-12 - 35\,x - 3\,x^{2} + 66\,x^{3} + 48\,x^{4} + 12\,n + 7\,n x - 189\,n x^{2} - 418\,n x^{3} - 240\,n x^{4} + 28\,n^{2} x + 252\,n^{2} x^{2} + 480\,n^{2} x^{3} + 204\,n^{2} x^{4} - 64\,n^{2} x^{5} - 60\,n^{3} x^{2} - 112\,n^{3} x^{3} + 60\,n^{3} x^{4} + 128\,n^{3} x^{5} - 16\,n^{4} x^{3} - 84\,n^{4} x^{4} - 80\,n^{4} x^{5} + 12\,n^{5} x^{4} + 16\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(2 + 18\,x + 38\,x^{2} - 6\,x^{3} - 76\,x^{4} - 48\,x^{5} + 56\,n x^{2} + 264\,n x^{3} + 400\,n x^{4} + 192\,n x^{5} - 28\,n^{2} x^{2} - 132\,n^{2} x^{3} - 168\,n^{2} x^{4} + 64\,n^{2} x^{6} - 32\,n^{3} x^{4} - 96\,n^{3} x^{5} - 64\,n^{3} x^{6} + 8\,n^{4} x^{4} + 24\,n^{4} x^{5} + 16\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
231619
313144216373
4631056324025606919
53217200334805376028000122761
61683472802948407168007560002903042106907
7898930340823738407741440123480009580032289766435253373
8486391916992180442087386624015800400018521395211300889628114944578217871

All 15 rows

Technical details
Node id
n-9b2d5bea1a9a453e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cdbe49d10d989431

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:       3,       16;
   3:      13,      144,       216;
   4:      63,     1056,      3240,       2560;
   5:     321,     7200,     33480,      53760,       28000;
   6:    1683,    47280,    294840,     716800,      756000,      290304;
   7:    8989,   303408,   2373840,    7741440,    12348000,     9580032,     2897664;
   8:   48639,  1916992,  18044208,   73866240,   158004000,   185213952,   113008896,    28114944;
   9:  265729, 11976768, 131770800,  649313280,  1743588000,  2739889152,  2523865344,  1265172480,   266872320;
  10: 1462563, 74195280, 934266960, 5383741440, 17405388000, 34331641344, 42378336000, 32051036160, 13610488320, 2489344000;
%F Formula
The row polynomials satisfy (4*n-4+(9*n-9)*x-(20*n^3-80*n^2+99*n-39)*x^2-(24*n^3-96*n^2+126*n-54)*x^3+(4*n^5-28*n^4+76*n^3-100*n^2+64*n-16)*x^4)*P(n,x) = (12*n-12+(28*n^2+7*n-35)*x-(60*n^3-252*n^2+189*n+3)*x^2-(16*n^4+112*n^3-480*n^2+418*n-66)*x^3+(12*n^5-84*n^4+60*n^3+204*n^2-240*n+48)*x^4+(16*n^5-80*n^4+128*n^3-64*n^2)*x^5)*P(n-1,x) + (2+18*x-(28*n^2-56*n-38)*x^2-(132*n^2-264*n+6)*x^3+(8*n^4-32*n^3-168*n^2+400*n-76)*x^4+(24*n^4-96*n^3+192*n-48)*x^5+(16*n^4-64*n^3+64*n^2)*x^6)*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[((12*n - 12 + (28*n^2 + 7*n - 35)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (16*n^4 + 112*n^3 - 480*n^2 + 418*n - 66)*x^3 + (12*n^5 - 84*n^4 + 60*n^3 + 204*n^2 - 240*n + 48)*x^4 + (16*n^5 - 80*n^4 + 128*n^3 - 64*n^2)*x^5)*P[n - 1, x] + (2 + 18*x - (28*n^2 - 56*n - 38)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (8*n^4 - 32*n^3 - 168*n^2 + 400*n - 76)*x^4 + (24*n^4 - 96*n^3 + 192*n - 48)*x^5 + (16*n^4 - 64*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(4*n - 4 + (9*n - 9)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (24*n^3 - 96*n^2 + 126*n - 54)*x^3 + (4*n^5 - 28*n^4 + 76*n^3 - 100*n^2 + 64*n - 16)*x^4)]];
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] = ((12*n - 12 + (28*n^2 + 7*n - 35)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (16*n^4 + 112*n^3 - 480*n^2 + 418*n - 66)*x^3 + (12*n^5 - 84*n^4 + 60*n^3 + 204*n^2 - 240*n + 48)*x^4 + (16*n^5 - 80*n^4 + 128*n^3 - 64*n^2)*x^5)*v[i - 1] + (2 + 18*x - (28*n^2 - 56*n - 38)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (8*n^4 - 32*n^3 - 168*n^2 + 400*n - 76)*x^4 + (24*n^4 - 96*n^3 + 192*n - 48)*x^5 + (16*n^4 - 64*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(4*n - 4 + (9*n - 9)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (24*n^3 - 96*n^2 + 126*n - 54)*x^3 + (4*n^5 - 28*n^4 + 76*n^3 - 100*n^2 + 64*n - 16)*x^4)); 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