Real-rooted atlas

A137320 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26201238
32410410832268
4120616852448802116
5720417669604960160019218608
65040321126124853280236005184448180912
740320277056586320589568322000981121568010241930080
83628802654784610804868696324371920165580837004845056230422440480
936288002799936069044832849844486089328027050688756403212963841244165120282591360
10399168003225081608434912321118948480882287200443701440146322624315494404285440332800112643833354880
11479001600402976512011089419264156766917121338481760074419065602791174848713044992122411520135168008673282457655742641920
126227020800543029529601562517457922333687769602131943580801288683475205364972166415649413120319570099244774400041012224221184053248865199059200
13871782912007851494246402350433299968368383419955235681429555202315823324672105178891417634171971891279950993408133699737601559207936120446976553779211468814279076403200
1413076743680001212595453440037611947773440615253351265286273172983168043300892963328211804214702087529884730368196813426867237821848064052802061312520861286434403532813647872245760249718562150400
Technical details
Node id
n-385c0b04bd494306 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-713e6aebbd821571

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 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:       2,        4;
  2:       6,       20,       12;
  3:      24,      104,      108,       32;
  4:     120,      616,      852,      448,       80;
  5:     720,     4176,     6960,     4960,     1600,      192;
  6:    5040,    32112,    61248,    53280,    23600,     5184,     448;
  7:   40320,   277056,   586320,   589568,   322000,    98112,   15680,    1024;
  8:  362880,  2654784,  6108048,  6869632,  4371920,  1655808,  370048,   45056,   2304;
  9: 3628800, 27999360, 69044832, 84984448, 60893280, 27050688, 7564032, 1296384, 124416, 5120;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+4*n*x+4*x^2)*P(n-2,x) + (2*n+1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n*x + 4*x^2)*P[n - 2, x] + (2*n + 1 + 4*x)*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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n*x + 4*x^2)*v[i - 2] + (2*n + 1 + 4*x)*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.