Real-rooted atlas

A109692 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 24\,x + 36\,x^{2} - 16\,n - 88\,n x - 120\,n x^{2} + 20\,n^{2} + 96\,n^{2} x + 112\,n^{2} x^{2} - 8\,n^{3} - 32\,n^{3} x - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2 + 12\,x - 12\,n - 32\,n x + 8\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
28403684
3484008164801744
438445441492819072840047328
53840592642719685489925131201814401578624
6460808801285219328145656322125472015723648465696062346496
7645120147179521073955843848555527709491208832783365421480961383782402842368000

All 15 rows

Technical details
Node id
n-612bffc48af75835 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-701ecfe9c165ea89

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:         8,         40,          36;
  3:        48,        400,         816,          480;
  4:       384,       4544,       14928,        19072,         8400;
  5:      3840,      59264,      271968,       548992,       513120,        181440;
  6:     46080,     880128,     5219328,     14565632,     21254720,      15723648,       4656960;
  7:    645120,   14717952,   107395584,    384855552,    770949120,     883278336,     542148096,     138378240;
  8:  10321920,  274194432,  2380637184,  10453512192,  26767269120,   41886621696,   39589111296,   20801986560,   4670265600;
  9: 185794560, 5637390336, 56835385344, 296085438464, 926085112320, 1846063770624, 2374106663936, 1912755511296, 879740766720, 176432256000;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(32*n^3-96*n^2+88*n-24)*x+(32*n^3-112*n^2+120*n-36)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(16*n^2-32*n+12)*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[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (32*n^3 - 96*n^2 + 88*n - 24)*x + (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*x)*P[n - 1, x])/(2*n - 3)]];
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] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (32*n^3 - 96*n^2 + 88*n - 24)*x + (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*x)*v[i - 1])/(2*n - 3)); 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.