Real-rooted atlas

A136124 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2125636104
360376432128996
436027364284230440010084
52520220324230031360100001152109364
6201601964164375444012801780003801631361274552
71814401929024482176851672322856000854784131712819215950152
818144002074176056898288688163844470760016773120364403242598420736213842304
91995840024267456071921908895928524870683480031326220885744512142049281306368512003062541312
102395008003071761920972266457614068646400114797840005794830720188170348839444480051632640384000012390446708933248
113113510400418489113601402175681282174618772481932017120001083945772804023696230410043252736166966272017740800010903552294912756376640640
124358914560061079284224021513660549123543017634816338439233440020739450124808583915992962457143377924879726156866063360005819770883008102469222412967225309440
136538371840009510605798400350190586137606079456605081661837815124800408562251102721852161319008059280441425921353923341056219582105600247169894401836417024809902081605632234721906663680
14104613949440001574003902464006031026639129601097225265217536117938806090560083179250932377640756553495257614323169583923236668135192832685633428480092686052659288213192704560632217621354905636864004474712882096640
Technical details
Node id
n-a2f31fda74d75dbc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b2b833f30f6f7f19

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