Real-rooted atlas

A245962 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-10 - 6\,x + 10\,n + n x - n x^{2} + 5\,n^{2} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-30 - 48\,x - 24\,x^{2} - 4\,x^{3} + 30\,n + 28\,n x + n x^{2} - 2\,n x^{3} + 20\,n^{2} x + 20\,n^{2} x^{2} + 4\,n^{2} x^{3} + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 75\,x - 84\,x^{2} - 42\,x^{3} - 8\,x^{4} - 25\,n x - 60\,n x^{2} - 47\,n x^{3} - 12\,n x^{4} - 5\,n^{2} x^{2} - 9\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2347
34610
4716629
511301556
61860458131
72911210528274
8472082409610601
976378513264451276
101236801065680175122735
11199121021451628550665798
1232221364230372815752881412293
13521374481908216416010149125936
14843652415624175841039531924411654619
Technical details
Node id
n-c71d0f6291ed6624 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-38943946daac5d2c

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,   4;
  3:  4,   6;
  4:  7,  16,   6;
  5: 11,  30,  15;
  6: 18,  60,  45,   8;
  7: 29, 112, 105,  28;
  8: 47, 208, 240,  96, 10;
  9: 76, 378, 513, 264, 45;
%F Formula
The row polynomials satisfy (10*n-10+(5*n^2+n-6)*x+(n^3-n)*x^2)*P(n,x) = (30*n-30+(20*n^2+28*n-48)*x+(3*n^3+20*n^2+n-24)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (25+(25*n+75)*x+(5*n^2+60*n+84)*x^2+(9*n^2+47*n+42)*x^3+(4*n^2+12*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((30*n - 30 + (20*n^2 + 28*n - 48)*x + (3*n^3 + 20*n^2 + n - 24)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (25 + (25*n + 75)*x + (5*n^2 + 60*n + 84)*x^2 + (9*n^2 + 47*n + 42)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 + n - 6)*x + (n^3 - n)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((30*n - 30 + (20*n^2 + 28*n - 48)*x + (3*n^3 + 20*n^2 + n - 24)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (25 + (25*n + 75)*x + (5*n^2 + 60*n + 84)*x^2 + (9*n^2 + 47*n + 42)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 + n - 6)*x + (n^3 - n)*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. [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.