Real-rooted atlas

A286724 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 6\,x + 16\,x^{2} + 4\,n x - 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 20\,x^{3} - 8\,n x^{3}\right)\,P_{n-1}'(x) + \left(4\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
318817
41187248139
51322887683841473
615080048009600384019091
71721800192008640013824046080291793
8198352858800470400169344022579206451205129307
911286272150528188160012042240361267204128768010321920101817089
101162103683386886096384609638403251404808360755208360755201857945602250495523
1112001620069120016934400243855360203212800092897280002090188800018579456000371589120054780588561
12124224200130680041817600819624960983549952070253568000281014272000562028544000449622835200817496064001455367098923
13128834848232320094089600240869376039341998080404660551680252912844800089924567040001618642206720011771943321600196199055360041888448785857
141338486723926208196310400636045696013568974848018996564787201709690830848094982823936000303945036595200497364605337600331576403558400510117543936001298019439099059
15139266248635980838476838415390735360415549854720759862591488093083167457280744665339658240372332669829120010831495849574400162472437743616009998303861145600142832912302080043074477771208913
Technical details
Node id
n-17e55966d5083866 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-31dfc9d837f82561

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: 1,   2;
   3: 1,   8,     8;
   4: 1,  18,    72,     48;
   5: 1,  32,   288,    768,     384;
   6: 1,  50,   800,   4800,    9600,     3840;
   7: 1,  72,  1800,  19200,   86400,   138240,     46080;
   8: 1,  98,  3528,  58800,  470400,  1693440,   2257920,    645120;
   9: 1, 128,  6272, 150528, 1881600, 12042240,  36126720,  41287680,  10321920;
  10: 1, 162, 10368, 338688, 6096384, 60963840, 325140480, 836075520, 836075520, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-6)*x+(4*n^2-16*n+16)*x^2)*P(n-1,x) - (4*x^2+(8*n-20)*x^3)*d/dx P(n-1,x) + 4*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 6)*x + (4*n^2 - 16*n + 16)*x^2)*P[n - 1, x] - (4*x^2 + (8*n - 20)*x^3)*D[P[n - 1, x], x] + 4*x^4*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 6)*x + (4*n^2 - 16*n + 16)*x^2)*v[i - 1] - (4*x^2 + (8*n - 20)*x^3)*deriv(v[i - 1], x) + 4*x^4*deriv(deriv(v[i - 1], x), 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

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.