Real-rooted atlas

A104684 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,x - 2\,x^{2} + 2\,n - 7\,n x + 6\,n x^{2} + 4\,n^{2} x - 6\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-22 + 9\,x + 38\,x^{2} + 8\,x^{3} + 14\,n - 57\,n x - 58\,n x^{2} + 8\,n x^{3} + 32\,n^{2} x - 2\,n^{2} x^{2} - 28\,n^{2} x^{3} + 14\,n^{3} x^{2} + 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2 + 16\,x^{2} + 18\,x^{3} + 4\,x^{4} - 8\,n x - 22\,n x^{2} - 10\,n x^{3} + 4\,n x^{4} - 4\,n^{2} x^{2} - 12\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
313321257
46322818032503
532115361872768804577
61683100201656010880280019242135
7898964032133740121600516009216448389625
848639403396101883611860807084002136962822410243608295

All 15 rows

Technical details
Node id
n-388eab6589a0653d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6f638175010ea5cb

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,        4;
   3:      13,       32,       12;
   4:      63,      228,      180,       32;
   5:     321,     1536,     1872,      768,       80;
   6:    1683,    10020,    16560,    10880,     2800,      192;
   7:    8989,    64032,   133740,   121600,    51600,     9216,      448;
   8:   48639,   403396,  1018836,  1186080,   708400,   213696,    28224,    1024;
   9:  265729,  2514944,  7452480, 10587136,  8187200,  3526656,   802816,   81920,   2304;
  10: 1462563, 15554628, 52906176, 88779264, 84450240, 47561472, 15654912, 2801664, 228096, 5120;
%F Formula
The row polynomials satisfy (2*n-2+(4*n^2-7*n+3)*x+(2*n^3-6*n^2+6*n-2)*x^2)*P(n,x) = (14*n-22+(32*n^2-57*n+9)*x+(14*n^3-2*n^2-58*n+38)*x^2+(12*n^3-28*n^2+8*n+8)*x^3)*P(n-1,x) - (2+8*n*x+(4*n^2+22*n-16)*x^2+(12*n^2+10*n-18)*x^3+(8*n^2-4*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((14*n - 22 + (32*n^2 - 57*n + 9)*x + (14*n^3 - 2*n^2 - 58*n + 38)*x^2 + (12*n^3 - 28*n^2 + 8*n + 8)*x^3)*P[n - 1, x] - (2 + 8*n*x + (4*n^2 + 22*n - 16)*x^2 + (12*n^2 + 10*n - 18)*x^3 + (8*n^2 - 4*n - 4)*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (4*n^2 - 7*n + 3)*x + (2*n^3 - 6*n^2 + 6*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((14*n - 22 + (32*n^2 - 57*n + 9)*x + (14*n^3 - 2*n^2 - 58*n + 38)*x^2 + (12*n^3 - 28*n^2 + 8*n + 8)*x^3)*v[i - 1] - (2 + 8*n*x + (4*n^2 + 22*n - 16)*x^2 + (12*n^2 + 10*n - 18)*x^3 + (8*n^2 - 4*n - 4)*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (4*n^2 - 7*n + 3)*x + (2*n^3 - 6*n^2 + 6*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, v[i])))};
T(10)

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