Real-rooted atlas

A114656 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 16\,x^{2} + 4\,n - 20\,n x + 56\,n x^{2} + 16\,n x^{3} - 20\,n^{2} x + 48\,n^{2} x^{2} - 64\,n^{2} x^{3} + 16\,n^{2} x^{4} - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} - 16\,n^{3} x^{4} + 48\,n^{4} x^{3} - 16\,n^{4} x^{4} + 16\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-4 - 40\,x + 16\,x^{2} - 208\,x^{3} - 64\,x^{5} + 20\,n + 32\,n x + 232\,n x^{2} - 8\,n x^{3} + 320\,n x^{4} + 96\,n x^{5} - 72\,n^{2} x + 48\,n^{2} x^{2} + 256\,n^{2} x^{3} - 112\,n^{2} x^{4} + 96\,n^{2} x^{5} - 168\,n^{3} x^{2} - 312\,n^{3} x^{3} - 592\,n^{3} x^{4} - 192\,n^{3} x^{5} + 208\,n^{4} x^{3} + 304\,n^{4} x^{4} - 32\,n^{4} x^{5} + 80\,n^{5} x^{4} + 96\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1 + 6\,x^{2} - 32\,x^{4} - 32\,x^{6} - 12\,n x + 10\,n x^{2} + 24\,n x^{3} - 48\,n x^{4} + 96\,n x^{5} + 32\,n x^{6} - 12\,n^{2} x^{2} - 8\,n^{2} x^{3} + 32\,n^{2} x^{4} + 32\,n^{2} x^{5} + 64\,n^{2} x^{6} + 48\,n^{3} x^{3} + 16\,n^{3} x^{4} - 192\,n^{3} x^{5} - 64\,n^{3} x^{6} + 16\,n^{4} x^{4} - 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
23811
3116436111
4454085401281121
519724325256307240011357
6903141204284042240140001152115255
74279810243180604505602580005529631361170355
8207934623922232468416192034860001314432197568819211883765

All 15 rows

Technical details
Node id
n-95116ccff4ef466d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d13aacdb3a9cb540

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,        8;
   3:     11,       64,       36;
   4:     45,      408,      540,       128;
   5:    197,     2432,     5256,      3072,       400;
   6:    903,    14120,    42840,     42240,     14000,      1152;
   7:   4279,    81024,   318060,    450560,    258000,     55296,      3136;
   8:  20793,   462392,  2232468,   4161920,   3486000,   1314432,    197568,     8192;
   9: 103049,  2632192, 15102864,  35080192,  39205600,  21934080,   5883136,   655360,   20736;
  10: 518859, 14969160, 99578160, 277670400, 390852000, 295094016, 118804224, 23887872, 2052864, 51200;
%F Formula
The row polynomials satisfy (4*n+4-(20*n^2+20*n)*x-(24*n^3-48*n^2-56*n+16)*x^2+(48*n^4-32*n^3-64*n^2+16*n)*x^3+(16*n^5-16*n^4-16*n^3+16*n^2)*x^4)*P(n,x) = (20*n-4-(72*n^2-32*n+40)*x-(168*n^3-48*n^2-232*n-16)*x^2+(208*n^4-312*n^3+256*n^2-8*n-208)*x^3+(80*n^5+304*n^4-592*n^3-112*n^2+320*n)*x^4+(96*n^5-32*n^4-192*n^3+96*n^2+96*n-64)*x^5)*P(n-1,x) + (1-12*n*x-(12*n^2-10*n-6)*x^2+(48*n^3-8*n^2+24*n)*x^3+(16*n^4+16*n^3+32*n^2-48*n-32)*x^4-(192*n^3-32*n^2-96*n)*x^5-(64*n^4+64*n^3-64*n^2-32*n+32)*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[((20*n - 4 - (72*n^2 - 32*n + 40)*x - (168*n^3 - 48*n^2 - 232*n - 16)*x^2 + (208*n^4 - 312*n^3 + 256*n^2 - 8*n - 208)*x^3 + (80*n^5 + 304*n^4 - 592*n^3 - 112*n^2 + 320*n)*x^4 + (96*n^5 - 32*n^4 - 192*n^3 + 96*n^2 + 96*n - 64)*x^5)*P[n - 1, x] + (1 - 12*n*x - (12*n^2 - 10*n - 6)*x^2 + (48*n^3 - 8*n^2 + 24*n)*x^3 + (16*n^4 + 16*n^3 + 32*n^2 - 48*n - 32)*x^4 - (192*n^3 - 32*n^2 - 96*n)*x^5 - (64*n^4 + 64*n^3 - 64*n^2 - 32*n + 32)*x^6)*D[P[n - 1, x], x])/(4*n + 4 - (20*n^2 + 20*n)*x - (24*n^3 - 48*n^2 - 56*n + 16)*x^2 + (48*n^4 - 32*n^3 - 64*n^2 + 16*n)*x^3 + (16*n^5 - 16*n^4 - 16*n^3 + 16*n^2)*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] = ((20*n - 4 - (72*n^2 - 32*n + 40)*x - (168*n^3 - 48*n^2 - 232*n - 16)*x^2 + (208*n^4 - 312*n^3 + 256*n^2 - 8*n - 208)*x^3 + (80*n^5 + 304*n^4 - 592*n^3 - 112*n^2 + 320*n)*x^4 + (96*n^5 - 32*n^4 - 192*n^3 + 96*n^2 + 96*n - 64)*x^5)*v[i - 1] + (1 - 12*n*x - (12*n^2 - 10*n - 6)*x^2 + (48*n^3 - 8*n^2 + 24*n)*x^3 + (16*n^4 + 16*n^3 + 32*n^2 - 48*n - 32)*x^4 - (192*n^3 - 32*n^2 - 96*n)*x^5 - (64*n^4 + 64*n^3 - 64*n^2 - 32*n + 32)*x^6)*deriv(v[i - 1], x))/(4*n + 4 - (20*n^2 + 20*n)*x - (24*n^3 - 48*n^2 - 56*n + 16)*x^2 + (48*n^4 - 32*n^3 - 64*n^2 + 16*n)*x^3 + (16*n^5 - 16*n^4 - 16*n^3 + 16*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, 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