Real-rooted atlas

A204420 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 72\,x + 108\,x^{2} + 108\,n - 240\,n x - 288\,n x^{2} - 144\,n^{2} + 224\,n^{2} x + 144\,n^{2} x^{2} + 80\,n^{3} - 64\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(-72\,x^{2} + 48\,x^{3} + 240\,n x^{2} - 128\,n x^{3} - 224\,n^{2} x^{2} + 64\,n^{2} x^{3} + 64\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(4 - 8\,x - 12\,n + 16\,n x + 8\,n^{2}\right)\,P_{n-1}(x) + \left(8\,x^{2} - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
2996108213
32252760486019209765
41102514784032508021504042000740985
58930251276884032319000278208009450000108864084340305
610805602516236158404542656580463034880020997900004311014403259872013468167405
7182614682252856289035608626904886609925006291205429724300001514722809602076538464011070259202875398611085

All 15 rows

Technical details
Node id
n-a1b8bba2b7403833 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-aed83f03f3b18f2e

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,                 8;
  2:                9,                96,               108;
  3:              225,              2760,              4860,              1920;
  4:            11025,            147840,            325080,            215040,             42000;
  5:           893025,          12768840,          32319000,          27820800,           9450000,           1088640;
  6:        108056025,        1623615840,        4542656580,        4630348800,        2099790000,         431101440,         32598720;
  7:      18261468225,      285628903560,      862690488660,      992500629120,      542972430000,      151472280960,      20765384640,      1107025920;
  8:    4108830350625,    66457879488000,   213385310938800,   269322800947200,   168692263740000,    57537672192000,   10857329683200,   1062744883200,    42032390400;
  9: 1187451971330625, 19765128099717000, 66752382642145200, 90732246046041600, 63059838021180000, 24887560496198400, 5800483885996800, 789287340441600, 57878601580800, 1764322560000;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(64*n^3-224*n^2+240*n-72)*x-(144*n^2-288*n+108)*x^2)*P(n-2,x) + ((64*n^3-224*n^2+240*n-72)*x^2+(64*n^2-128*n+48)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+4+(16*n-8)*x)*P(n-1,x) - (16*n-8)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (64*n^3 - 224*n^2 + 240*n - 72)*x - (144*n^2 - 288*n + 108)*x^2)*P[n - 2, x] + ((64*n^3 - 224*n^2 + 240*n - 72)*x^2 + (64*n^2 - 128*n + 48)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 4 + (16*n - 8)*x)*P[n - 1, x] - (16*n - 8)*x^2*D[P[n - 1, x], 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (64*n^3 - 224*n^2 + 240*n - 72)*x - (144*n^2 - 288*n + 108)*x^2)*v[i - 2] + ((64*n^3 - 224*n^2 + 240*n - 72)*x^2 + (64*n^2 - 128*n + 48)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 4 + (16*n - 8)*x)*v[i - 1] - (16*n - 8)*x^2*deriv(v[i - 1], x)); 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.