Real-rooted atlas

A103327 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n x + 4\,n^{2} - 4\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(6\,x^{2} - 8\,n - 36\,n x - 26\,n x^{2} + 16\,n^{2} + 32\,n^{2} x + 16\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x^{2} + 4\,x^{3} - 8\,n x - 24\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
216481276
36432028832704
4256179228801280806288
5102492162150417920480019254656
64096450561382401720328960016128448465600
716384212992811008135168010752003870725017610243905536

All 15 rows

Technical details
Node id
n-ae385526595d7f42 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d88a651a3889a86b

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,       12;
  3:     64,     320,      288,       32;
  4:    256,    1792,     2880,     1280,       80;
  5:   1024,    9216,    21504,    17920,     4800,      192;
  6:   4096,   45056,   138240,   172032,    89600,    16128,      448;
  7:  16384,  212992,   811008,  1351680,  1075200,   387072,    50176,    1024;
  8:  65536,  983040,  4472832,  9371648, 10137600,  5677056,  1505280,  147456,   2304;
  9: 262144, 4456448, 23592960, 59637760, 82001920, 63258624, 26492928, 5406720, 414720, 5120;
%F Formula
The row polynomials satisfy (4*n^2-2*n+(4*n^3-4*n^2+n)*x)*P(n,x) = (16*n^2-8*n+(16*n^3+32*n^2-36*n)*x+(24*n^3-26*n+6)*x^2)*P(n-1,x) - (8*n*x+(8*n^2+24*n-2)*x^2+(16*n^2+16*n-4)*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[((16*n^2 - 8*n + (16*n^3 + 32*n^2 - 36*n)*x + (24*n^3 - 26*n + 6)*x^2)*P[n - 1, x] - (8*n*x + (8*n^2 + 24*n - 2)*x^2 + (16*n^2 + 16*n - 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 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] = ((16*n^2 - 8*n + (16*n^3 + 32*n^2 - 36*n)*x + (24*n^3 - 26*n + 6)*x^2)*v[i - 1] - (8*n*x + (8*n^2 + 24*n - 2)*x^2 + (16*n^2 + 16*n - 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 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