Real-rooted atlas

A204420 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 48\,x + 27\,x^{2} + 176\,n - 152\,n x - 72\,n x^{2} - 204\,n^{2} + 128\,n^{2} x + 36\,n^{2} x^{2} + 96\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(-48\,x^{2} + 12\,x^{3} + 152\,n x^{2} - 32\,n x^{3} - 128\,n^{2} x^{2} + 16\,n^{2} x^{3} + 32\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(6 - 4\,x - 16\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x) + \left(4\,x^{2} - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1044
20242751
304808102401530
40201604158020160262584525
50145152034020002116800472500340207476840
601596672004101451202993760008835750011226600509355969281775
702490808320068652904320561815654401986484500034054020002781078308648640173299556430
805230697472000152577577152001362887285760054883728900001144215072000127929601800726485760016418902540885274655225
901422749712384000435018427683840041681767615488001854854361360000445581867931200612727171056004816600588800200967366600344594250012307840711065900
100486580401635328000154858557288314880015724560102451200007581029391936000002031378141592800003247851144578040031678411564800001845550316610005892561675000792222180754604699617534291275
110204363768686837760000673396864588191744000718245385690587955200369999447297480000000108242914828112064000194473306619471880002221332456805862400161708391534690000725963598360000018300332375325019799007228002096086197220705339650
1201034080669555399065600003513146385111688765440003909666147337102344192002130316638807803846400006702529659751724198400013228953898592007576000170820791160062883840014648604560138629080082650955673286000002946353512427325006010978594420800534435702050251140838494329851402810725
13062044840173323943936000000216605486872950445670400000250193952774055868497920000143091131716503786816000000478843178582184390374400001021809423726516625200000014568905885470551014400001419403933138204213200009480231587353527000000426800351657329650000123725976068494800002084299237995975001549547301802500731646574228694614540050000
1404355547780167340864307200000015582627582534064245473280000018603321821728884107771904000011100503184764114029708800000039178174337607011435688960000893285083584967434352992000013830814144252205849548800001494268391868083343111600001138646901811618999800000060933301282228514139000022399611183179390160000538374493174360342500761447544105748500048028060502296875546075555866691865501546434375
Technical details
Node id
n-7df447b935390019 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-751d7e1b87540300

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: 0,                4;
  2: 0,               24,               27;
  3: 0,              480,              810,              240;
  4: 0,            20160,            41580,            20160,             2625;
  5: 0,          1451520,          3402000,          2116800,           472500,           34020;
  6: 0,        159667200,        410145120,        299376000,         88357500,        11226600,         509355;
  7: 0,      24908083200,      68652904320,      56181565440,      19864845000,      3405402000,      278107830,       8648640;
  8: 0,    5230697472000,   15257757715200,   13628872857600,    5488372890000,   1144215072000,   127929601800,    7264857600,    164189025;
  9: 0, 1422749712384000, 4350184276838400, 4168176761548800, 1854854361360000, 445581867931200, 61272717105600, 4816600588800, 200967366600, 3445942500;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(32*n^3-128*n^2+152*n-48)*x-(36*n^2-72*n+27)*x^2)*P(n-2,x) + ((32*n^3-128*n^2+152*n-48)*x^2+(16*n^2-32*n+12)*x^3)*d/dx P(n-2,x) + (8*n^2-16*n+6+(8*n-4)*x)*P(n-1,x) - (8*n-4)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (32*n^3 - 128*n^2 + 152*n - 48)*x - (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + ((32*n^3 - 128*n^2 + 152*n - 48)*x^2 + (16*n^2 - 32*n + 12)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 16*n + 6 + (8*n - 4)*x)*P[n - 1, x] - (8*n - 4)*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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (32*n^3 - 128*n^2 + 152*n - 48)*x - (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + ((32*n^3 - 128*n^2 + 152*n - 48)*x^2 + (16*n^2 - 32*n + 12)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 16*n + 6 + (8*n - 4)*x)*v[i - 1] - (8*n - 4)*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

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.