Real-rooted atlas

A327997 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-54 + 45\,n - 9\,n^{2}\right)\,P_{n-4}(x) + \left(-18 - 24\,x - 3\,n + 12\,n x + 6\,n^{2}\right)\,P_{n-3}(x) + \left(24\,x^{2} - 12\,n x^{2}\right)\,P_{n-3}'(x) + \left(5 - 8\,x + 9\,x^{2} - 10\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(8\,x^{2} + 4\,x^{3} + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(5 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)
\(P_{2}(x) = 17 + 36\,x + 9\,x^{2}\)
\(P_{3}(x) = 78 + 260\,x + 135\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
21736962
37826013516489
439318081449352254027
52208129161404052007503635150
613977979561336866648014500140449328052
71000268029801304541808864235375342722401643288523
8806769717580813349250982060835484756864487203038408135463309
9728060470065252145027584122166560522774001252465217410681386245832100411227676
10728650897462238761682575659158083568077468262521956583637736517397248024907585001215118715458
118016931268636637060208842354272145794833611747290500381372512477493269710074451283397604235001197914468225982165
129620848953108130940496277170968043306829177248184172875150668763025921556958903523922328322443816981636800068812716416169971024388759
13125072630712145765950958839261976294324629529083504300328413385011979356772243122419259965498471264066375454865483478003038406810764002197019614714122678866
14175102161293721063395506500592154258432407371556340366451106727337575220814598644646331422330233124926922348817289741397516878061200115319304153818128163017428812225236705269267656
Technical details
Node id
n-e6b6a28c50cf2da7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6d262060b387a414

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:       4,        4;
  2:      17,       36,         9;
  3:      78,      260,       135,        16;
  4:     393,     1808,      1449,       352,       25;
  5:    2208,    12916,     14040,      5200,      750,       36;
  6:   13977,    97956,    133686,     66480,    14500,     1404,      49;
  7:  100026,   802980,   1304541,    808864,   235375,    34272,    2401,     64;
  8:  806769,  7175808,  13349250,   9820608,  3548475,   686448,   72030,   3840,   81;
  9: 7280604, 70065252, 145027584, 122166560, 52277400, 12524652, 1741068, 138624, 5832, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-3*n-18+(12*n-24)*x)*P(n-3,x) - (12*n-24)*x^2*d/dx P(n-3,x) - (n^2+10*n-5+(4*n+8)*x-9*x^2)*P(n-2,x) + ((4*n+8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n+5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 4*x, P(2,x) = 17 + 36*x + 9*x^2, P(3,x) = 78 + 260*x + 135*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[2, x_] := 17 + 36*x + 9*x^2;
P[3, x_] := 78 + 260*x + 135*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 3*n - 18 + (12*n - 24)*x)*P[n - 3, x] - (12*n - 24)*x^2*D[P[n - 3, x], x] - (n^2 + 10*n - 5 + (4*n + 8)*x - 9*x^2)*P[n - 2, x] + ((4*n + 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n + 5 + 4*x)*P[n - 1, x] - 4*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] = 4 + 4*x; v[3] = 17 + 36*x + 9*x^2; v[4] = 78 + 260*x + 135*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 3*n - 18 + (12*n - 24)*x)*v[i - 3] - (12*n - 24)*x^2*deriv(v[i - 3], x) - (n^2 + 10*n - 5 + (4*n + 8)*x - 9*x^2)*v[i - 2] + ((4*n + 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n + 5 + 4*x)*v[i - 1] - 4*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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023