Real-rooted atlas

A046716 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-16 + 8\,n\right)\,P_{n}(x) = \left(264 + 3456\,x + 14976\,x^{2} + 21504\,x^{3} - 232\,n - 3168\,n x - 14208\,n x^{2} - 20992\,n x^{3} + 72\,n^{2} + 960\,n^{2} x + 4224\,n^{2} x^{2} + 6144\,n^{2} x^{3} - 8\,n^{3} - 96\,n^{3} x - 384\,n^{3} x^{2} - 512\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 408\,x + 2592\,x^{2} + 7296\,x^{3} + 7680\,x^{4} - 23\,n - 400\,n x - 2592\,n x^{2} - 7424\,n x^{3} - 7936\,n x^{4} + 8\,n^{2} + 136\,n^{2} x + 864\,n^{2} x^{2} + 2432\,n^{2} x^{3} + 2560\,n^{2} x^{4} - n^{3} - 16\,n^{3} x - 96\,n^{3} x^{2} - 256\,n^{3} x^{3} - 256\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-120 - 896\,x - 1536\,x^{2} + 104\,n + 832\,n x + 1536\,n x^{2} - 24\,n^{2} - 192\,n^{2} x - 384\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8 - 136\,x - 736\,x^{2} - 1280\,x^{3} + 13\,n + 192\,n x + 944\,n x^{2} + 1536\,n x^{3} - 6\,n^{2} - 84\,n^{2} x - 384\,n^{2} x^{2} - 576\,n^{2} x^{3} + n^{3} + 12\,n^{3} x + 48\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(40 + 256\,x - 48\,n - 256\,n x + 16\,n^{2} + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 72\,x - 160\,x^{2} + 5\,n + 48\,n x + 112\,n x^{2} - n^{2} - 8\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 8\,x\)
\(P_{3}(x) = 5 + 40\,x + 48\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
22810
35404893
4162005282561000
565115251367168128014801
63267688505921308161203206144315886
7195758504520512204057637171202586624286728953965
8137005002005670288302581768449152011978342468210688131072309059068

All 15 rows

Technical details
Node id
n-cd3498cf408c35bf · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5b45f722c8e5d293

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,        8;
   3:      5,       40,        48;
   4:     16,      200,       528,        256;
   5:     65,     1152,      5136,       7168,        1280;
   6:    326,     7688,     50592,     130816,      120320,        6144;
   7:   1957,    58504,    520512,    2040576,     3717120,     2586624,        28672;
   8:  13700,   500200,   5670288,   30258176,    84491520,   119783424,     68210688,       131072;
   9: 109601,  4749888,  65826720,  448002048,  1704728320,  3711811584,   4333772800,   2107637760,      589824;
  10: 986410, 49634952, 816118848, 6785721856, 32901127680, 97505593344, 174603419648, 173946961920, 74130259968, 2621440;
%F Formula
The row polynomials satisfy (8*n-16)*P(n,x) = -(8*n^3-72*n^2+232*n-264+(96*n^3-960*n^2+3168*n-3456)*x+(384*n^3-4224*n^2+14208*n-14976)*x^2+(512*n^3-6144*n^2+20992*n-21504)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(16*n^3-136*n^2+400*n-408)*x+(96*n^3-864*n^2+2592*n-2592)*x^2+(256*n^3-2432*n^2+7424*n-7296)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*d/dx P(n-3,x) - (24*n^2-104*n+120+(192*n^2-832*n+896)*x+(384*n^2-1536*n+1536)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(12*n^3-84*n^2+192*n-136)*x+(48*n^3-384*n^2+944*n-736)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*d/dx P(n-2,x) + (16*n^2-48*n+40+(64*n^2-256*n+256)*x)*P(n-1,x) - (n^2-5*n+8+(8*n^2-48*n+72)*x+(16*n^2-112*n+160)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 8*x, P(3,x) = 5 + 40*x + 48*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[3, x_] := 5 + 40*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 72*n^2 + 232*n - 264 + (96*n^3 - 960*n^2 + 3168*n - 3456)*x + (384*n^3 - 4224*n^2 + 14208*n - 14976)*x^2 + (512*n^3 - 6144*n^2 + 20992*n - 21504)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*D[P[n - 3, x], x] - (24*n^2 - 104*n + 120 + (192*n^2 - 832*n + 896)*x + (384*n^2 - 1536*n + 1536)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*D[P[n - 2, x], x] + (16*n^2 - 48*n + 40 + (64*n^2 - 256*n + 256)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];
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 + 8*x; v[3] = 5 + 40*x + 48*x^2; for(n = 4, N, my(i = n); v[i] = (-(8*n^3 - 72*n^2 + 232*n - 264 + (96*n^3 - 960*n^2 + 3168*n - 3456)*x + (384*n^3 - 4224*n^2 + 14208*n - 14976)*x^2 + (512*n^3 - 6144*n^2 + 20992*n - 21504)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*deriv(v[i - 3], x) - (24*n^2 - 104*n + 120 + (192*n^2 - 832*n + 896)*x + (384*n^2 - 1536*n + 1536)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*deriv(v[i - 2], x) + (16*n^2 - 48*n + 40 + (64*n^2 - 256*n + 256)*x)*v[i - 1] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*deriv(v[i - 1], x))/(8*n - 16)); 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