Real-rooted atlas

A304236 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(26\,n + 19\,n x + 13\,n^{2} x + 3\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-156 - 348\,x - 228\,x^{2} - 36\,x^{3} + 182\,n + 250\,n x + 54\,n x^{2} - 18\,n x^{3} + 130\,n^{2} x + 159\,n^{2} x^{2} + 36\,n^{2} x^{3} + 21\,n^{3} x^{2} + 18\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 572\,x - 709\,x^{2} - 378\,x^{3} - 72\,x^{4} - 169\,n x - 442\,n x^{2} - 381\,n x^{3} - 108\,n x^{4} - 39\,n^{2} x^{2} - 75\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
24610
371219
4195427100
54013281253
6974084051081018
7217104412424322935

All 15 rows

Technical details
Node id
n-3d868bb9b7a407a7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8126c1326009a233

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:    4,    6;
  3:    7,   12;
  4:   19,   54,    27;
  5:   40,  132,    81;
  6:   97,  408,   405,  108;
  7:  217, 1044,  1242,  432;
  8:  508, 2850,  4293, 2376,  405;
  9: 1159, 7284, 12717, 8640, 2025;
%F Formula
The row polynomials satisfy (26*n+(13*n^2+19*n)*x+(3*n^3+3*n^2)*x^2)*P(n,x) = (182*n-156+(130*n^2+250*n-348)*x+(21*n^3+159*n^2+54*n-228)*x^2+(18*n^3+36*n^2-18*n-36)*x^3)*P(n-1,x) - (169+(169*n+572)*x+(39*n^2+442*n+709)*x^2+(75*n^2+381*n+378)*x^3+(36*n^2+108*n+72)*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[((182*n - 156 + (130*n^2 + 250*n - 348)*x + (21*n^3 + 159*n^2 + 54*n - 228)*x^2 + (18*n^3 + 36*n^2 - 18*n - 36)*x^3)*P[n - 1, x] - (169 + (169*n + 572)*x + (39*n^2 + 442*n + 709)*x^2 + (75*n^2 + 381*n + 378)*x^3 + (36*n^2 + 108*n + 72)*x^4)*D[P[n - 1, x], x])/(26*n + (13*n^2 + 19*n)*x + (3*n^3 + 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] = ((182*n - 156 + (130*n^2 + 250*n - 348)*x + (21*n^3 + 159*n^2 + 54*n - 228)*x^2 + (18*n^3 + 36*n^2 - 18*n - 36)*x^3)*v[i - 1] - (169 + (169*n + 572)*x + (39*n^2 + 442*n + 709)*x^2 + (75*n^2 + 381*n + 378)*x^3 + (36*n^2 + 108*n + 72)*x^4)*deriv(v[i - 1], x))/(26*n + (13*n^2 + 19*n)*x + (3*n^3 + 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

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.