Real-rooted atlas

A204420 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 54\,x - 27\,x^{2} + 108\,n - 180\,n x + 72\,n x^{2} - 144\,n^{2} + 168\,n^{2} x - 36\,n^{2} x^{2} + 80\,n^{3} - 48\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 6\,x - 12\,n + 12\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
297281162
32252070364516207560
41102511088024381018144042525589680
5893025957663024239250234738009568125137781069128640
610805602512177118803406992435390685680021260373755456127605304568511364312960
7182614682252142216776706470178664958374224058205497595853751917071055903379010134523643219602494544532480

All 15 rows

Technical details
Node id
n-321747bdedd90149 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e496ecd3ecc24bd1

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:                1,                 6;
  2:                9,                72,                81;
  3:              225,              2070,              3645,              1620;
  4:            11025,            110880,            243810,            181440,             42525;
  5:           893025,           9576630,          24239250,          23473800,           9568125,           1377810;
  6:        108056025,        1217711880,        3406992435,        3906856800,        2126037375,         545612760,         53045685;
  7:      18261468225,      214221677670,      647017866495,      837422405820,      549759585375,      191707105590,      33790101345,       2364321960;
  8:    4108830350625,    49843409616000,   160038983204100,   227241113299200,   170800917036750,    72821116368000,   17667395846100,    2269749081600,    119693799225;
  9: 1187451971330625, 14823846074787750, 50064286981608900, 76555332601347600, 63848085996444750, 31498318753001100, 9438733823418900, 1685714271040800, 164818361532825, 6782648622750;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 6)*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] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 6)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.