Real-rooted atlas

A204420 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 36\,x - 12\,x^{2} + 108\,n - 120\,n x + 32\,n x^{2} - 144\,n^{2} + 112\,n^{2} x - 16\,n^{2} x^{2} + 80\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 4\,x - 12\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
29483693
3225138016204803705
41102573920108360537608400255465
58930256384420107730006955200189000018144027077085
6108056025811807920151421886011575872004199580007185024046569604088135205
718261468225142814451780287563496220248125157280108594486000252453801602966483520138378240833709301425

All 15 rows

Technical details
Node id
n-c6ff1e446479ba16 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3f072313d028fa7f

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,                4;
  2:                9,               48,                36;
  3:              225,             1380,              1620,               480;
  4:            11025,            73920,            108360,             53760,              8400;
  5:           893025,          6384420,          10773000,           6955200,           1890000,           181440;
  6:        108056025,        811807920,        1514218860,        1157587200,         419958000,         71850240,         4656960;
  7:      18261468225,     142814451780,      287563496220,      248125157280,      108594486000,      25245380160,      2966483520,      138378240;
  8:    4108830350625,   33228939744000,    71128436979600,    67330700236800,    33738452748000,    9589612032000,   1551047097600,   132843110400,    4670265600;
  9: 1187451971330625, 9882564049858500, 22250794214048400, 22683061511510400, 12611967604236000, 4147926749366400, 828640555142400, 98660917555200, 6430955731200, 176432256000;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(32*n^3-112*n^2+120*n-36)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (8*n - 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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (8*n - 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

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.