Real-rooted atlas

A143495 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-11 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(-3 - 4\,x - x^{2}\right)\,P_{n-2}'(x) + \left(7 + 2\,x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2527
32620349
4141158484351
579911685379252601
64736851252501380155620039
72937162532484081728029952407160833
819049746821843608619748446760579635081345199
9129102035942563911295215087664991511037610318488911718553
109131275283759103531178222870852843979518351902353681723265710106218071
116731084723076566432304711324093752010528287528056672461535946380027360860111000508081
1251636983819345442403007243734253842362412837182604065416528213297410595376857835416901100129780470335
13411641679716719948344285605496362691349672015475483870569213462413697959662172780722258902815061206139113801399089261961
143405116498514895367458627718371049228837714536818590641498078018568056218597298924136252120529061247452923902531386878281703141039063635047
15291871399682136723812388427516642109353131843958152223781559183010564131898323385727264927471381174411409650109120196076086227141409936812257720721511262835074593
Technical details
Node id
n-65b757298c3a21a2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a6e4670daa2b48fb

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:       5,        2;
   3:      26,       20,        3;
   4:     141,      158,       48,        4;
   5:     799,     1168,      537,       92,       5;
   6:    4736,     8512,     5250,     1380,     155,       6;
   7:   29371,    62532,    48408,    17280,    2995,     240,      7;
   8:  190497,   468218,   436086,   197484,   46760,    5796,    350,     8;
   9: 1291020,  3594256,  3911295,  2150876,  649915,  110376,  10318,   488,   9;
  10: 9131275, 28375910, 35311782, 22870852, 8439795, 1835190, 235368, 17232, 657, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(11+6*x+x^2)*P(n-2,x) - (3+4*x+x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(11 + 6*x + x^2)*P[n - 2, x] - (3 + 4*x + x^2)*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 2*x; for(n = 3, N, my(i = n); v[i] = -(11 + 6*x + x^2)*v[i - 2] - (3 + 4*x + x^2)*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); 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

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.