Real-rooted atlas

A104684 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 42\,x - 180\,x^{2} + 192\,x^{3} - 64\,x^{4} + 4\,n - 82\,n x + 500\,n x^{2} - 800\,n x^{3} + 320\,n x^{4} + 40\,n^{2} x - 464\,n^{2} x^{2} + 1216\,n^{2} x^{3} - 640\,n^{2} x^{4} + 144\,n^{3} x^{2} - 800\,n^{3} x^{3} + 640\,n^{3} x^{4} + 192\,n^{4} x^{3} - 320\,n^{4} x^{4} + 64\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-44 + 366\,x + 420\,x^{2} - 896\,x^{3} + 64\,x^{4} + 28\,n - 766\,n x + 1292\,n x^{2} + 4448\,n x^{3} - 2880\,n x^{4} - 1024\,n x^{5} + 336\,n^{2} x - 2912\,n^{2} x^{2} - 2208\,n^{2} x^{3} + 9536\,n^{2} x^{4} + 1280\,n^{2} x^{5} + 1200\,n^{3} x^{2} - 2560\,n^{3} x^{3} - 8256\,n^{3} x^{4} + 1280\,n^{3} x^{5} + 1472\,n^{4} x^{3} + 1344\,n^{4} x^{4} - 2304\,n^{4} x^{5} + 448\,n^{5} x^{4} + 768\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-1 + 18\,x + 96\,x^{2} - 48\,x^{3} - 320\,x^{4} - 24\,n x + 28\,n x^{2} + 728\,n x^{3} + 640\,n x^{4} - 1280\,n x^{5} - 512\,n x^{6} - 120\,n^{2} x^{2} - 400\,n^{2} x^{3} + 1024\,n^{2} x^{4} + 2944\,n^{2} x^{5} + 512\,n^{2} x^{6} - 192\,n^{3} x^{3} - 1088\,n^{3} x^{4} - 1152\,n^{3} x^{5} + 512\,n^{3} x^{6} - 64\,n^{4} x^{4} - 384\,n^{4} x^{5} - 512\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
231619
313128144285
463912216010244159
532161442246424576640059905
616834008019872034816022400036864849507
78989256128160488038912004128000176947220070411859373
848639161358412226032379545605667200041029632126443521048576163237375

All 15 rows

Technical details
Node id
n-3f87f13d0f813f00 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fcf88ff765517704

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,      128,       144;
   4:      63,      912,      2160,       1024;
   5:     321,     6144,     22464,      24576,       6400;
   6:    1683,    40080,    198720,     348160,     224000,      36864;
   7:    8989,   256128,   1604880,    3891200,    4128000,    1769472,     200704;
   8:   48639,  1613584,  12226032,   37954560,   56672000,   41029632,   12644352,    1048576;
   9:  265729, 10059776,  89429760,  338788352,  654976000,  677117952,  359661568,   83886080,   5308416;
  10: 1462563, 62218512, 634874112, 2840936448, 6756019200, 9131802624, 7013400576, 2868903936, 525533184, 26214400;
%F Formula
The row polynomials satisfy (4*n-4+(40*n^2-82*n+42)*x+(144*n^3-464*n^2+500*n-180)*x^2+(192*n^4-800*n^3+1216*n^2-800*n+192)*x^3+(64*n^5-320*n^4+640*n^3-640*n^2+320*n-64)*x^4)*P(n,x) = (28*n-44+(336*n^2-766*n+366)*x+(1200*n^3-2912*n^2+1292*n+420)*x^2+(1472*n^4-2560*n^3-2208*n^2+4448*n-896)*x^3+(448*n^5+1344*n^4-8256*n^3+9536*n^2-2880*n+64)*x^4+(768*n^5-2304*n^4+1280*n^3+1280*n^2-1024*n)*x^5)*P(n-1,x) - (1+(24*n-18)*x+(120*n^2-28*n-96)*x^2+(192*n^3+400*n^2-728*n+48)*x^3+(64*n^4+1088*n^3-1024*n^2-640*n+320)*x^4+(384*n^4+1152*n^3-2944*n^2+1280*n)*x^5+(512*n^4-512*n^3-512*n^2+512*n)*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[((28*n - 44 + (336*n^2 - 766*n + 366)*x + (1200*n^3 - 2912*n^2 + 1292*n + 420)*x^2 + (1472*n^4 - 2560*n^3 - 2208*n^2 + 4448*n - 896)*x^3 + (448*n^5 + 1344*n^4 - 8256*n^3 + 9536*n^2 - 2880*n + 64)*x^4 + (768*n^5 - 2304*n^4 + 1280*n^3 + 1280*n^2 - 1024*n)*x^5)*P[n - 1, x] - (1 + (24*n - 18)*x + (120*n^2 - 28*n - 96)*x^2 + (192*n^3 + 400*n^2 - 728*n + 48)*x^3 + (64*n^4 + 1088*n^3 - 1024*n^2 - 640*n + 320)*x^4 + (384*n^4 + 1152*n^3 - 2944*n^2 + 1280*n)*x^5 + (512*n^4 - 512*n^3 - 512*n^2 + 512*n)*x^6)*D[P[n - 1, x], x])/(4*n - 4 + (40*n^2 - 82*n + 42)*x + (144*n^3 - 464*n^2 + 500*n - 180)*x^2 + (192*n^4 - 800*n^3 + 1216*n^2 - 800*n + 192)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*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] = ((28*n - 44 + (336*n^2 - 766*n + 366)*x + (1200*n^3 - 2912*n^2 + 1292*n + 420)*x^2 + (1472*n^4 - 2560*n^3 - 2208*n^2 + 4448*n - 896)*x^3 + (448*n^5 + 1344*n^4 - 8256*n^3 + 9536*n^2 - 2880*n + 64)*x^4 + (768*n^5 - 2304*n^4 + 1280*n^3 + 1280*n^2 - 1024*n)*x^5)*v[i - 1] - (1 + (24*n - 18)*x + (120*n^2 - 28*n - 96)*x^2 + (192*n^3 + 400*n^2 - 728*n + 48)*x^3 + (64*n^4 + 1088*n^3 - 1024*n^2 - 640*n + 320)*x^4 + (384*n^4 + 1152*n^3 - 2944*n^2 + 1280*n)*x^5 + (512*n^4 - 512*n^3 - 512*n^2 + 512*n)*x^6)*deriv(v[i - 1], x))/(4*n - 4 + (40*n^2 - 82*n + 42)*x + (144*n^3 - 464*n^2 + 500*n - 180)*x^2 + (192*n^4 - 800*n^3 + 1216*n^2 - 800*n + 192)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*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