Real-rooted atlas

A110441 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16\,n + 4\,n x + 4\,n x^{2} - 20\,n^{3} x^{2} - 10\,n^{3} x^{3} + n^{5} x^{4}\right)\,P_{n}(x) = \left(32 + 44\,x + 28\,x^{2} + 4\,x^{3} - 4\,x^{4} + 32\,n + 72\,n x + 24\,n x^{2} - 30\,n x^{3} - 22\,n x^{4} + 28\,n^{2} x - 44\,n^{2} x^{2} - 84\,n^{2} x^{3} - 36\,n^{2} x^{4} + n^{2} x^{5} - 40\,n^{3} x^{2} - 54\,n^{3} x^{3} - 18\,n^{3} x^{4} + 3\,n^{3} x^{5} - 4\,n^{4} x^{3} + 2\,n^{4} x^{4} + 3\,n^{4} x^{5} + 2\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(8 + 24\,x + 20\,x^{2} - 8\,x^{4} - 2\,x^{5} - 28\,n x^{2} - 76\,n x^{3} - 54\,n x^{4} - 10\,n x^{5} - 28\,n^{2} x^{2} - 76\,n^{2} x^{3} - 52\,n^{2} x^{4} - 6\,n^{2} x^{5} + n^{2} x^{6} + 4\,n^{3} x^{4} + 8\,n^{3} x^{5} + 2\,n^{3} x^{6} + 2\,n^{4} x^{4} + 4\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2146436114
348352432128960
41641664324020484007516
5560726419584194568000115256016
619123020810454414336092000276483136402808
765281216005149449062408000003732488780881922818560

All 15 rows

Technical details
Node id
n-83492a54f04fd664 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7862e7a130f7088d

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:     4,       8;
  2:    14,      64,       36;
  3:    48,     352,      432,      128;
  4:   164,    1664,     3240,     2048,      400;
  5:   560,    7264,    19584,    19456,     8000,     1152;
  6:  1912,   30208,   104544,   143360,    92000,    27648,     3136;
  7:  6528,  121600,   514944,   906240,   800000,   373248,    87808,    8192;
  8: 22288,  478208,  2397888,  5169152,  5848000,  3741696,  1361024,  262144,  20736;
  9: 76096, 1847936, 10713600, 27389952, 37945600, 31062528, 15454208, 4587520, 746496, 51200;
%F Formula
The row polynomials satisfy (16*n+4*n*x-(20*n^3-4*n)*x^2-10*n^3*x^3+n^5*x^4)*P(n,x) = (32*n+32+(28*n^2+72*n+44)*x-(40*n^3+44*n^2-24*n-28)*x^2-(4*n^4+54*n^3+84*n^2+30*n-4)*x^3+(2*n^5+2*n^4-18*n^3-36*n^2-22*n-4)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (8+24*x-(28*n^2+28*n-20)*x^2-(76*n^2+76*n)*x^3+(2*n^4+4*n^3-52*n^2-54*n-8)*x^4+(4*n^4+8*n^3-6*n^2-10*n-2)*x^5+(n^4+2*n^3+n^2)*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[((32*n + 32 + (28*n^2 + 72*n + 44)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (4*n^4 + 54*n^3 + 84*n^2 + 30*n - 4)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (8 + 24*x - (28*n^2 + 28*n - 20)*x^2 - (76*n^2 + 76*n)*x^3 + (2*n^4 + 4*n^3 - 52*n^2 - 54*n - 8)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(16*n + 4*n*x - (20*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*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] = ((32*n + 32 + (28*n^2 + 72*n + 44)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (4*n^4 + 54*n^3 + 84*n^2 + 30*n - 4)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (8 + 24*x - (28*n^2 + 28*n - 20)*x^2 - (76*n^2 + 76*n)*x^3 + (2*n^4 + 4*n^3 - 52*n^2 - 54*n - 8)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(16*n + 4*n*x - (20*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*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