Real-rooted atlas

A143491 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 24\,x + 36\,x^{2} - 6\,n - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(24\,x^{2} + 16\,x^{3} + 8\,n x^{2}\right)\,P_{n-2}'(x) + \left(7 + 8\,x + 2\,n\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 12 + 56\,x + 36\,x^{2}\)
\(P_{1}(x) = 60 + 376\,x + 432\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0125636104
160376432128996
236027364284230440010084
32520220324230031360100001152109364
4201601964164375444012801780003801631361274552
51814401929024482176851672322856000854784131712819215950152
618144002074176056898288688163844470760016773120364403242598420736213842304
71995840024267456071921908895928524870683480031326220885744512142049281306368512003062541312
82395008003071761920972266457614068646400114797840005794830720188170348839444480051632640384000012390446708933248
93113510400418489113601402175681282174618772481932017120001083945772804023696230410043252736166966272017740800010903552294912756376640640
104358914560061079284224021513660549123543017634816338439233440020739450124808583915992962457143377924879726156866063360005819770883008102469222412967225309440
116538371840009510605798400350190586137606079456605081661837815124800408562251102721852161319008059280441425921353923341056219582105600247169894401836417024809902081605632234721906663680
12104613949440001574003902464006031026639129601097225265217536117938806090560083179250932377640756553495257614323169583923236668135192832685633428480092686052659288213192704560632217621354905636864004474712882096640
131778437140480002759497793740800109610470426291202079719453594419223478425989200000175371102739123209192938147352960349959981853081698591352837120020709628825600032348957921280370570552934430236344320016634347520552960000838860889633041354575360
14320118685286400051093709999718400209716586839157760413322113354121216487602900730425600383285851779317760213212797953559040870070025062318082660480555133772861620875796480001083454260162560143698566316032141406563450881000758312960481443840001409286144189399041882451386960128000
Technical details
Node id
n-6e5e3c059ded51eb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b2b833f30f6f7f19

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,          56,           36;
  1:         60,         376,          432,          128;
  2:        360,        2736,         4284,         2304,          400;
  3:       2520,       22032,        42300,        31360,        10000,         1152;
  4:      20160,      196416,       437544,       401280,       178000,        38016,        3136;
  5:     181440,     1929024,      4821768,      5167232,      2856000,       854784,      131712,        8192;
  6:    1814400,    20741760,     56898288,     68816384,     44707600,     16773120,     3644032,      425984,      20736;
  7:   19958400,   242674560,    719219088,    959285248,    706834800,    313262208,    85744512,    14204928,    1306368,     51200;
  8:  239500800,  3071761920,   9722664576,  14068646400,  11479784000,   5794830720,  1881703488,   394444800,   51632640,   3840000,   123904;
  9: 3113510400, 41848911360, 140217568128, 217461877248, 193201712000, 108394577280, 40236962304, 10043252736, 1669662720, 177408000, 10903552, 294912;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(8*n+24)*x-36*x^2)*P(n-2,x) + ((8*n+24)*x^2+16*x^3)*d/dx P(n-2,x) + (2*n+7+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 56*x + 36*x^2, P(1,x) = 60 + 376*x + 432*x^2 + 128*x^3.
%t Mathematica
P[0, x_] := 12 + 56*x + 36*x^2;
P[1, x_] := 60 + 376*x + 432*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (8*n + 24)*x - 36*x^2)*P[n - 2, x] + ((8*n + 24)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n + 7 + 8*x)*P[n - 1, x] - 8*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 + 56*x + 36*x^2; v[2] = 60 + 376*x + 432*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (8*n + 24)*x - 36*x^2)*v[i - 2] + ((8*n + 24)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n + 7 + 8*x)*v[i - 1] - 8*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.