Real-rooted atlas

A088996 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 24\,n x - 16\,n x^{2} + 12\,n^{2} + 40\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2206484
313680011522088
411929952272642457662984
5128281372485439369482246144002256636
616378821254721092422427836416352051201769472093949740
7242030436803264231284736769711104143194112014129233925780275204463111440

All 15 rows

Technical details
Node id
n-86d5927b033e7262 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4e2a9fa5562b05a2

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:         0;
  1:         4;
  2:        20,          64;
  3:       136,         800,         1152;
  4:      1192,        9952,        27264,        24576;
  5:     12828,      137248,       543936,       948224,        614400;
  6:    163788,     2125472,     10924224,     27836416,      35205120,      17694720;
  7:   2420304,    36803264,    231284736,    769711104,    1431941120,    1412923392,     578027520;
  8:  40628880,   706948416,   5235827712,  21413815296,   52267560960,   76180389888,   61419552768,   21139292160;
  9: 763663860, 14943402432, 127184285568, 615354590208, 1852170224640, 3553025261568, 4243708772352, 2886343852032, 856141332480;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(16*n^3-40*n^2+24*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 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.