Real-rooted atlas

A103327 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-n + n x + 2\,n^{2} - 4\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(12\,x^{2} - 4\,n - 36\,n x - 52\,n x^{2} + 8\,n^{2} + 32\,n^{2} x + 16\,n^{3} x + 48\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x^{2} + 8\,x^{3} - 4\,n x - 24\,n x^{2} - 32\,n x^{3} - 8\,n^{2} x^{2} - 32\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2169648160
36464011522562112
425635841152010240128026880
510241843286016143360768006144331776
640969011255296013762561433600516096286724001792
7163844259843244032108134401720320012386304321126413107247431680

All 15 rows

Technical details
Node id
n-9f682d82235ea46e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f86d77f8169a378b

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:     16,      96,       48;
  3:     64,     640,     1152,       256;
  4:    256,    3584,    11520,     10240,       1280;
  5:   1024,   18432,    86016,    143360,      76800,       6144;
  6:   4096,   90112,   552960,   1376256,    1433600,     516096,      28672;
  7:  16384,  425984,  3244032,  10813440,   17203200,   12386304,    3211264,    131072;
  8:  65536, 1966080, 17891328,  74973184,  162201600,  181665792,   96337920,  18874368,    589824;
  9: 262144, 8912896, 94371840, 477102080, 1312030720, 2024275968, 1695547392, 692060160, 106168320, 2621440;
%F Formula
The row polynomials satisfy (2*n^2-n+(4*n^3-4*n^2+n)*x)*P(n,x) = (8*n^2-4*n+(16*n^3+32*n^2-36*n)*x+(48*n^3-52*n+12)*x^2)*P(n-1,x) - (4*n*x+(8*n^2+24*n-2)*x^2+(32*n^2+32*n-8)*x^3)*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[((8*n^2 - 4*n + (16*n^3 + 32*n^2 - 36*n)*x + (48*n^3 - 52*n + 12)*x^2)*P[n - 1, x] - (4*n*x + (8*n^2 + 24*n - 2)*x^2 + (32*n^2 + 32*n - 8)*x^3)*D[P[n - 1, x], x])/(2*n^2 - n + (4*n^3 - 4*n^2 + n)*x)]];
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] = ((8*n^2 - 4*n + (16*n^3 + 32*n^2 - 36*n)*x + (48*n^3 - 52*n + 12)*x^2)*v[i - 1] - (4*n*x + (8*n^2 + 24*n - 2)*x^2 + (32*n^2 + 32*n - 8)*x^3)*deriv(v[i - 1], x))/(2*n^2 - n + (4*n^3 - 4*n^2 + n)*x)); 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

Same fingerprint ?

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