Real-rooted atlas

A207823 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + 8\,n^{2} x - 24\,n^{2} x^{2} + 16\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-8\,x - 52\,x^{2} - 96\,x^{3} + 2\,n + 8\,n x - 68\,n x^{2} - 288\,n x^{3} - 144\,n x^{4} + 4\,n^{2} + 56\,n^{2} x + 192\,n^{2} x^{2} + 144\,n^{2} x^{3} + 16\,n^{3} x + 120\,n^{3} x^{2} + 240\,n^{3} x^{3} + 144\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 12\,x^{2} - 44\,x^{3} - 48\,x^{4} - 2\,n x - 32\,n x^{2} - 164\,n x^{3} - 288\,n x^{4} - 144\,n x^{5} - 8\,n^{2} x^{2} - 76\,n^{2} x^{3} - 192\,n^{2} x^{4} - 144\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
21164562
3124138224387
413227992810452285
514046823685455468013012
6148705480016830294722037772233
715699084804005010670415027686912393469

All 15 rows

Technical details
Node id
n-9db049b57ec83477 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c5d07ab8e49bbf20

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: 1,  8;
  2: 1, 16,   45;
  3: 1, 24,  138,   224;
  4: 1, 32,  279,   928,   1045;
  5: 1, 40,  468,  2368,   5455,   4680;
  6: 1, 48,  705,  4800,  16830,  29472,   20377;
  7: 1, 56,  990,  8480,  40050, 106704,  150276,   86912;
  8: 1, 64, 1323, 13664,  81275, 291744,  624666,  734592,  364905;
  9: 1, 72, 1704, 20608, 147945, 669144, 1930068, 3447936, 3477357, 1513160;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(16*n^3+8*n^2)*x+(24*n^3-24*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(16*n^3+56*n^2+8*n-8)*x+(120*n^3+192*n^2-68*n-52)*x^2+(240*n^3+144*n^2-288*n-96)*x^3+(144*n^3-144*n)*x^4)*P(n-1,x) - ((2*n+1)*x+(8*n^2+32*n+12)*x^2+(76*n^2+164*n+44)*x^3+(192*n^2+288*n+48)*x^4+(144*n^2+144*n)*x^5)*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[((4*n^2 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (120*n^3 + 192*n^2 - 68*n - 52)*x^2 + (240*n^3 + 144*n^2 - 288*n - 96)*x^3 + (144*n^3 - 144*n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (76*n^2 + 164*n + 44)*x^3 + (192*n^2 + 288*n + 48)*x^4 + (144*n^2 + 144*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (24*n^3 - 24*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] = ((4*n^2 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (120*n^3 + 192*n^2 - 68*n - 52)*x^2 + (240*n^3 + 144*n^2 - 288*n - 96)*x^3 + (144*n^3 - 144*n)*x^4)*v[i - 1] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (76*n^2 + 164*n + 44)*x^3 + (192*n^2 + 288*n + 48)*x^4 + (144*n^2 + 144*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (24*n^3 - 24*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023