Real-rooted atlas

A145324 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 16\,n x - 3\,n^{2} - 32\,n^{2} x + 144\,n^{2} x^{2} - n^{3} - 16\,n^{3} x + 144\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(16\,n x^{2} + 32\,n^{2} x^{2} + 64\,n^{2} x^{3} + 16\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 4\,n + 16\,n x + 2\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-16\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 32\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133235
2122728641148
3602128141122457640876
4360175681804326492167680001615576
52520157536221169612242944297216002654208070878376
620160154252827618336208036864773401600141071155210115481603432879200
718144016463232359627040344350822417362457600483347496967054825881642278584320182343830368

All 15 rows

Technical details
Node id
n-8fc38d95be59e819 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3476278f7f0e1eea

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:        3,         32;
  2:       12,        272,         864;
  3:       60,       2128,       14112,        24576;
  4:      360,      17568,      180432,       649216,        768000;
  5:     2520,     157536,     2211696,     12242944,      29721600,       26542080;
  6:    20160,    1542528,    27618336,    208036864,     773401600,     1410711552,      1011548160;
  7:   181440,   16463232,   359627040,   3443508224,   17362457600,    48334749696,     70548258816,     42278584320;
  8:  1814400,  190759680,  4929792192,  57451212800,  367321913600,  1383417298944,   3073885323264,   3739993767936,   1926317998080;
  9: 19958400, 2388660480, 71396085312, 982526341120, 7631241849600, 36375332511744, 109132124831744, 201734756499456, 210526682480640, 95126814720000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(16*n^3+32*n^2+16*n)*x-(144*n^3+144*n^2)*x^2)*P(n-2,x) + ((16*n^3+32*n^2+16*n)*x^2+(64*n^3+64*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(16*n^2+16*n)*x)*P(n-1,x) - (16*n^2+16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 32*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (16*n^3 + 32*n^2 + 16*n)*x - (144*n^3 + 144*n^2)*x^2)*P[n - 2, x] + ((16*n^3 + 32*n^2 + 16*n)*x^2 + (64*n^3 + 64*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (16*n^2 + 16*n)*x)*P[n - 1, x] - (16*n^2 + 16*n)*x^2*D[P[n - 1, x], x])/(n)]];
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] = 3 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (16*n^3 + 32*n^2 + 16*n)*x - (144*n^3 + 144*n^2)*x^2)*v[i - 2] + ((16*n^3 + 32*n^2 + 16*n)*x^2 + (64*n^3 + 64*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (16*n^2 + 16*n)*x)*v[i - 1] - (16*n^2 + 16*n)*x^2*deriv(v[i - 1], x))/(n)); 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.