Real-rooted atlas

A145324 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 8\,n x - 3\,n^{2} - 16\,n^{2} x + 36\,n^{2} x^{2} - n^{3} - 8\,n^{3} x + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,n x^{2} + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 4\,n + 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131619
212136216364
3601064352830727724
43608784451088115248000183404
5252078768552924153036818576008294404851620
620160771264690458426004608483376004408473615805440141928392
71814408231616899067604304385281085153600151046092811023165443303014404556990856
818144009537984012324480487181401600229576196004323179059248029458176292187013127524679680159473293248
9199584001194330240178490213281228157926404769526156001136729140992170518945049615760527851528223698534401857945600006044967508288
10239500800160883020802733076028162171884568064994522425920028750608554112545009960167686791073028505653751657977856245655060480004945851187200246832094301952
113113510400232140003840442176714892839895623771648210733586672000717464861601408164769949431897625912788790476802761559986120704191199329648640077767836106752014126331985920010804224429608704
1243589145600357376513536075484930309632762922636575744457102992913120017934373547419392484580771893599369223243407292825612393111041935257611541057237196800071038806162407424260407260138700804310493246259200504769647873862400
136538371840005848846133760013574211581721601520131163516006410194331428715680045331952995386086414103700608142714243155667619733372928512745353874968371260151969704460288004975692284398387200275778829885951180892038458964967424013997625405603840025074410336039840000
1410461394944000101427584348160025666709671042560315616781265076224234365451149263680011657063656466947584410764684461443280641057541618496809287682018560115567464427522861488381474951168002979627265775650406402217691637619260129281118232831234067660803426246850951249920048206107901952000001319816759158806464000
Technical details
Node id
n-3ea9bc8be3781e4d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c32d5c1e1e574210

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,         16;
  2:       12,        136,         216;
  3:       60,       1064,        3528,         3072;
  4:      360,       8784,       45108,        81152,        48000;
  5:     2520,      78768,      552924,      1530368,      1857600,        829440;
  6:    20160,     771264,     6904584,     26004608,     48337600,      44084736,      15805440;
  7:   181440,    8231616,    89906760,    430438528,   1085153600,    1510460928,    1102316544,     330301440;
  8:  1814400,   95379840,  1232448048,   7181401600,  22957619600,   43231790592,   48029458176,   29218701312,   7524679680;
  9: 19958400, 1194330240, 17849021328, 122815792640, 476952615600, 1136729140992, 1705189450496, 1576052785152, 822369853440, 185794560000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(8*n^3+16*n^2+8*n)*x-(36*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^3+16*n^2+8*n)*x^2+(16*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(8*n^2+8*n)*x)*P(n-1,x) - (8*n^2+8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x - (36*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^3 + 16*n^2 + 8*n)*x^2 + (16*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*P[n - 1, x] - (8*n^2 + 8*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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x - (36*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^3 + 16*n^2 + 8*n)*x^2 + (16*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*v[i - 1] - (8*n^2 + 8*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.