Real-rooted atlas

A128100 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(36\,n + 13\,n x + 9\,n^{2} x + n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-144 - 160\,x - 52\,x^{2} - 4\,x^{3} + 180\,n + 119\,n x + 12\,n x^{2} - 2\,n x^{3} + 63\,n^{2} x + 37\,n^{2} x^{2} + 4\,n^{2} x^{3} + 5\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-324 - 540\,x - 329\,x^{2} - 86\,x^{3} - 8\,x^{4} - 162\,n x - 207\,n x^{2} - 87\,n x^{3} - 12\,n x^{4} - 18\,n^{2} x^{2} - 17\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2325
3549
41114328
52132962
64382364165
78518810216391

All 15 rows

Technical details
Node id
n-6b9f88f14eacd57d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9913b280f2bd95b5

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:   3,   2;
  3:   5,   4;
  4:  11,  14,   3;
  5:  21,  32,   9;
  6:  43,  82,  36,   4;
  7:  85, 188, 102,  16;
  8: 171, 438, 297,  72,  5;
  9: 341, 984, 783, 240, 25;
%F Formula
The row polynomials satisfy (36*n+(9*n^2+13*n)*x+(n^3+n^2)*x^2)*P(n,x) = (180*n-144+(63*n^2+119*n-160)*x+(5*n^3+37*n^2+12*n-52)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (324+(162*n+540)*x+(18*n^2+207*n+329)*x^2+(17*n^2+87*n+86)*x^3+(4*n^2+12*n+8)*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[((180*n - 144 + (63*n^2 + 119*n - 160)*x + (5*n^3 + 37*n^2 + 12*n - 52)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (324 + (162*n + 540)*x + (18*n^2 + 207*n + 329)*x^2 + (17*n^2 + 87*n + 86)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(36*n + (9*n^2 + 13*n)*x + (n^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] = ((180*n - 144 + (63*n^2 + 119*n - 160)*x + (5*n^3 + 37*n^2 + 12*n - 52)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (324 + (162*n + 540)*x + (18*n^2 + 207*n + 329)*x^2 + (17*n^2 + 87*n + 86)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(36*n + (9*n^2 + 13*n)*x + (n^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.