Real-rooted atlas

A136394 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 + 6\,x - 3\,x^{2} + 4\,n - 8\,n x + 4\,n x^{2} - n^{2} + 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(4 - 4\,x - 6\,n + 6\,n x + 2\,n^{2} - 2\,n^{2} x\right)\,P_{n-3}(x) + \left(1 - 2\,x + 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)
\(P_{3}(x) = 1 + 10\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2123
311011
4140950
51168105274
61818885601764
7147307077126013068
81321285901017920525109584
91251328531234226688173251026576
101222414652149782813720370125567010628640
1112195234655822338357671606731725270270120543840
1212389625686492017074747413441153979758066520727651486442880
1312843085256816489636366355889601955900375198558360472972519802759040
14136696680226110528696967979794375563359697841544751606901876124251081080283465647360
Technical details
Node id
n-97a6d6300e173698 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-949d153525c7ff77

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: 1,      2;
  3: 1,     10;
  4: 1,     40,      9;
  5: 1,    168,    105;
  6: 1,    818,    885,     60;
  7: 1,   4730,   7077,   1260;
  8: 1,  32128,  59010,  17920,   525;
  9: 1, 251328, 531234, 226688, 17325;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-4*n+3-(2*n^2-8*n+6)*x+(n^2-4*n+3)*x^2)*P(n-4,x) + (2*n^2-6*n+4-(2*n^2-6*n+4)*x)*P(n-3,x) - (n^2-1-(2*n-2)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 10*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 1 + 10*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 4, x] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*P[n - 3, x] - (n^2 - 1 - (2*n - 2)*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x]];
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; v[3] = 1 + 2*x; v[4] = 1 + 10*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*v[i - 4] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*v[i - 3] - (n^2 - 1 - (2*n - 2)*x)*v[i - 2] + (2*n - 1)*v[i - 1]); 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.