Real-rooted atlas

A103327 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2\,n x - 16\,n^{2} + 28\,n^{2} x - n^{2} x^{2} + 32\,n^{3} - 72\,n^{3} x + 6\,n^{3} x^{2} + 48\,n^{4} x - 12\,n^{4} x^{2} + 8\,n^{5} x^{2}\right)\,P_{n}(x) = \left(12\,x^{2} - 3\,x^{3} - 8\,n x + 48\,n x^{2} + 14\,n x^{3} - 64\,n^{2} - 208\,n^{2} x - 152\,n^{2} x^{2} - 13\,n^{2} x^{3} + 128\,n^{3} + 96\,n^{3} x - 120\,n^{3} x^{2} - 38\,n^{3} x^{3} + 192\,n^{4} x + 192\,n^{4} x^{2} + 20\,n^{4} x^{3} + 32\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(4\,x^{2} + 2\,x^{3} - 2\,x^{4} + 24\,n x^{2} + 32\,n x^{3} + 8\,n x^{4} - 64\,n^{2} x - 112\,n^{2} x^{2} - 52\,n^{2} x^{3} - 4\,n^{2} x^{4} - 96\,n^{3} x^{2} - 128\,n^{3} x^{3} - 32\,n^{3} x^{4} - 16\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
21648973
36432021616616
425617922160640254873
51024921616128896015003636864
64096450561036808601628000302449269921
716384212992608256675840336000725765488641927600

All 15 rows

Technical details
Node id
n-566cd3f241c2cf50 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e31b485f3411f36c

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,       4;
  2:     16,      48,        9;
  3:     64,     320,      216,       16;
  4:    256,    1792,     2160,      640,       25;
  5:   1024,    9216,    16128,     8960,     1500,       36;
  6:   4096,   45056,   103680,    86016,    28000,     3024,      49;
  7:  16384,  212992,   608256,   675840,   336000,    72576,    5488,     64;
  8:  65536,  983040,  3354624,  4685824,  3168000,  1064448,  164640,   9216,    81;
  9: 262144, 4456448, 17694720, 29818880, 25625600, 11860992, 2897664, 337920, 14580, 100;
%F Formula
The row polynomials satisfy (32*n^3-16*n^2+(48*n^4-72*n^3+28*n^2-2*n)*x+(8*n^5-12*n^4+6*n^3-n^2)*x^2)*P(n,x) = (128*n^3-64*n^2+(192*n^4+96*n^3-208*n^2-8*n)*x+(32*n^5+192*n^4-120*n^3-152*n^2+48*n+12)*x^2+(24*n^5+20*n^4-38*n^3-13*n^2+14*n-3)*x^3)*P(n-1,x) - (64*n^2*x+(96*n^3+112*n^2-24*n-4)*x^2+(16*n^4+128*n^3+52*n^2-32*n-2)*x^3+(16*n^4+32*n^3+4*n^2-8*n+2)*x^4)*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[((128*n^3 - 64*n^2 + (192*n^4 + 96*n^3 - 208*n^2 - 8*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*P[n - 1, x] - (64*n^2*x + (96*n^3 + 112*n^2 - 24*n - 4)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*D[P[n - 1, x], x])/(32*n^3 - 16*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (8*n^5 - 12*n^4 + 6*n^3 - n^2)*x^2)]];
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] = ((128*n^3 - 64*n^2 + (192*n^4 + 96*n^3 - 208*n^2 - 8*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*v[i - 1] - (64*n^2*x + (96*n^3 + 112*n^2 - 24*n - 4)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*deriv(v[i - 1], x))/(32*n^3 - 16*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (8*n^5 - 12*n^4 + 6*n^3 - n^2)*x^2)); 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