Real-rooted atlas

A145324 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 6\,n x - 3\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,n + 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131215
212102162276
360798264625926096
43606588338316847248600157851
52520590764146931291248188082010497604698117
620160578448517843821941388489418205579474425719120158174118
71814406173712674300703631825081098718020191167711217937249127054387205946526494
818144007153488092433603660593076002324458984554715234968781550781486240361305621427701120247003210053
91995840089574768013386765996103625825040482914523295143867281906827747390835083366042422976234182665296071425670400011236380502923
102395008001206622656020498070211218325276043041006953956244036387488951298886857725362141450398116537281530662447885049443805772320025927518355200555664247604360
11311351040017410500288033163253616963366193255732821336775650540090804146546428226811895789252985534303621559840786397355422653673503445419417604076804824456320101824435722240029683425176734140
124358914560026803238515205661369773222464371597461078446281678032453402269819152095266878852540783579003196984700505365328352913201155109484443677010937001200372405048391689984187704935350709760430208240926464001703587674125639295
1365383718400043866346003200101806586862912012826106692166304103217605715746260573732530097855156229499949628036577767396924651922720641460122511620515400823124437884626793920260839536311648052001987849624747989657691858869198469900801945864966652006400104545395556005868065
141046139494400076070688261120019250032253281920266301659192408064237295019288629476014753471190215980536668409497706232481222258636249660665929845748165329096412373681100052261964286403120156200293338766318224015985409363986778732161116050882782196799360476296051735940083200938184894635788800006831695407621499820090
Technical details
Node id
n-a9896c2cd46e63b5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-677b43bbbe9b90a7

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,        12;
  2:       12,       102,         162;
  3:       60,       798,        2646,         2592;
  4:      360,      6588,       33831,        68472,        48600;
  5:     2520,     59076,      414693,      1291248,      1880820,       1049760;
  6:    20160,    578448,     5178438,     21941388,     48941820,      55794744,      25719120;
  7:   181440,   6173712,    67430070,    363182508,   1098718020,    1911677112,    1793724912,     705438720;
  8:  1814400,  71534880,   924336036,   6059307600,  23244589845,   54715234968,   78155078148,   62403613056,   21427701120;
  9: 19958400, 895747680, 13386765996, 103625825040, 482914523295, 1438672819068, 2774739083508, 3366042422976, 2341826652960, 714256704000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(6*n^3+12*n^2+6*n)*x+(9*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(6*n^2+6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*P[n - 1, x])/(n)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*v[i - 1])/(n)); 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.