Real-rooted atlas

A059364 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 4\,n x - 4\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
2221638
3244376144764
4303672885808153617668
5428461408881732089433619200470478
6681738285984847876883997696166464027648014268090
7121179606198638413187212814933561694929280321131524515840486870360
8238437720144154324837294993925351977536460051008023685657606762147848257536018489323880
951505863303597307963210978215614419119555859220782918608014437376716862587961856154796359681672151040774044082810
101212568944309614581254723384617667984694263089427291443018724808019749576448468306843289617556543406083834069811203715891200035433303697710
1130913934045402744884024077610957432450420825896262882912040124129265952042586380969177631355081308544015812577043251252270805495808102268256256008992456704001761255749639700
12848744545509008346078894373203727516361273136998044410486265617799911181361600222023444708885761976295699850572812553395134013440557616593907302416495587871948802924693382758402354388664320094487788546080300
13249716246597055026947884651454440133192649442430152398692558022093760804962035304799520115482052957247347212070709545766081289261838194311045125177432880248647682056273254941491205507405344722124889311788977356806631528071168005442406592138022150
14783945368200984509213297529865295604994397989191560888165346097618247780483729813855461246768060536519743842994560727291949820891024646548560130995349094444189345664800870400220707614001038950407930469175189209088194126618424036556829014385797103616019996607722291200335020169521348690050
Technical details
Node id
n-7938474c51929009 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9783f43730ed991f

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:          2;
  2:         22,          16;
  3:        244,         376,          144;
  4:       3036,        7288,         5808,         1536;
  5:      42846,      140888,       173208,        94336,        19200;
  6:     681738,     2859848,      4787688,      3997696,      1664640,       276480;
  7:   12117960,    61986384,    131872128,    149335616,     94929280,     32113152,     4515840;
  8:  238437720,  1441543248,   3729499392,   5351977536,   4600510080,   2368565760,   676214784,    82575360;
  9: 5150586330, 35973079632, 109782156144, 191195558592, 207829186080, 144373767168, 62587961856, 15479635968, 1672151040;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(8*n^3-12*n^2+4*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*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. [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.