Real-rooted atlas

A143491 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 48\,x + 144\,x^{2} - 6\,n - 16\,n x - n^{2}\right)\,P_{n-2}(x) + \left(48\,x^{2} + 64\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(7 + 16\,x + 2\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 12 + 112\,x + 144\,x^{2}\)
\(P_{1}(x) = 60 + 752\,x + 1728\,x^{2} + 1024\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
012112144268
160752172810243564
236054721713618432640047800
325204406416920025088016000036864663528
42016039283217501763210240284800012165122007049638624
518144038580481928707241337856456960002735308884295681048576147191648
61814400414835202275931525505310727153216005367398402332180485452595253084162366536000
719958400485349120287687635276742819841130935680010024390656548764876818182307843344302082621440040056737472
823950080061435238403889065830411254917120018367654400018543458304012042902323250488934400132179558401966080000126877696713162852352
931135104008369782272056087027251217396950179843091227392000346862647296025751655874561285536350208427433656320908328960001116523724860397977613337968195584
1043589145600122158568448086054642196482834414107852854150277350400663662403993605493706235494431451435237376124920989614083382444032000595944538112616059371522835349504261654724288512
116538371840001902121159680014007623445504048635652840652898940504199680013073992035287041185383244165120758789650251776346604375310336112426038067200253101971865603760982065152331735891968131533373445375531433443328
121046139494400031480078049280024124106556518408777802121740288188702089744896002661736029836083226084194236964864183336570674216969387042609364992351044315381760094910517923020818066061865779222963495632896174939386675260397977600115472920378613760
13177843714048000551899558748160043844188170516480166377556287553536375654815827200000561187528765194240588348041430589440447948776771944448252393863263027200106033299587072000331253329113907207589284924096512123848066334720013626857488384090596966400002748779069442589583611052892160
1432011868528640001021874199994368008388663473566310403306576906832969728780164641168680960012265147256938168320136456190690277785601113689632079767142468108302211424583683154988840779776000110945716240646144029429466381523353657920128389480448819821209976832078879758745600046179488366592124124554854460536666366446878720
Technical details
Node id
n-9196ed29b60d5d3c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dcd95a62876f0c72

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         12,         112,          144;
  1:         60,         752,         1728,          1024;
  2:        360,        5472,        17136,         18432,          6400;
  3:       2520,       44064,       169200,        250880,        160000,         36864;
  4:      20160,      392832,      1750176,       3210240,       2848000,       1216512,        200704;
  5:     181440,     3858048,     19287072,      41337856,      45696000,      27353088,       8429568,       1048576;
  6:    1814400,    41483520,    227593152,     550531072,     715321600,     536739840,     233218048,      54525952,      5308416;
  7:   19958400,   485349120,   2876876352,    7674281984,   11309356800,   10024390656,    5487648768,    1818230784,    334430208,    26214400;
  8:  239500800,  6143523840,  38890658304,  112549171200,  183676544000,  185434583040,  120429023232,   50488934400,  13217955840,  1966080000,   126877696;
  9: 3113510400, 83697822720, 560870272512, 1739695017984, 3091227392000, 3468626472960, 2575165587456, 1285536350208, 427433656320, 90832896000, 11165237248, 603979776;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(16*n+48)*x-144*x^2)*P(n-2,x) + ((16*n+48)*x^2+64*x^3)*d/dx P(n-2,x) + (2*n+7+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 112*x + 144*x^2, P(1,x) = 60 + 752*x + 1728*x^2 + 1024*x^3.
%t Mathematica
P[0, x_] := 12 + 112*x + 144*x^2;
P[1, x_] := 60 + 752*x + 1728*x^2 + 1024*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (16*n + 48)*x - 144*x^2)*P[n - 2, x] + ((16*n + 48)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (2*n + 7 + 16*x)*P[n - 1, x] - 16*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] = 12 + 112*x + 144*x^2; v[2] = 60 + 752*x + 1728*x^2 + 1024*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (16*n + 48)*x - 144*x^2)*v[i - 2] + ((16*n + 48)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (2*n + 7 + 16*x)*v[i - 1] - 16*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

Same fingerprint ?

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.