Real-rooted atlas

A078812 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(60 + 54\,x - 54\,x^{2} - 60\,n - 54\,n x + 162\,n x^{2} - 162\,n^{2} x^{2} + 54\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-90\,n - 171\,n x - 81\,n x^{2} - 45\,n^{2} x - 162\,n^{2} x^{2} - 81\,n^{2} x^{3} + 81\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 135\,x - 207\,x^{2} - 81\,x^{3} - 45\,n x^{2} - 162\,n x^{3} - 81\,n x^{4} + 45\,n^{2} x^{2} + 162\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2369
38362771
421150243108522
555540137712964053673
61441800631892886075145825083
73775724256775184052650262445103167615
89871763496471248940346275266814107163174961101780
92584531003430891080864191970020207881250235419904590497149313
10676515717011717464366008946606512706470107163005528736159432319683045910413
111771145900038794141670414442851025701764567537131052907904233834045904900649539292304807
124636813260961253199661264080181727550352086588461015226411610896247120065954625502143478721257641847751966
131213933797496396840062172160807320719251640302488253997742627625484162107163565110421630037868123776527504690873311609216569
14317811107947501236087547491201842829048525720427980612899975760165845283841539797153410241064900475852271414667771602694405872232052272557771391
1583204030493620379718631252457948810564274340301571116206137492917591219525056100059534333809455501804769240107519897151040556843879893746192471744535451423745855
Technical details
Node id
n-d9ef4a2b315ba571 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-84bb372e2b53dd93

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    3,      6;
   3:    8,     36,      27;
   4:   21,    150,     243,     108;
   5:   55,    540,    1377,    1296,     405;
   6:  144,   1800,    6318,    9288,    6075,     1458;
   7:  377,   5724,   25677,   51840,   52650,    26244,     5103;
   8:  987,  17634,   96471,  248940,  346275,   266814,   107163,   17496;
   9: 2584,  53100,  343089, 1080864, 1919700,  2020788,  1250235,  419904,   59049;
  10: 6765, 157170, 1171746, 4366008, 9466065, 12706470, 10716300, 5528736, 1594323, 196830;
%F Formula
The row polynomials satisfy -(60*n-60+(54*n-54)*x-(54*n^3-162*n^2+162*n-54)*x^2)*P(n,x) = -(90*n+(45*n^2+171*n)*x-(81*n^3-162*n^2-81*n)*x^2-(81*n^3-81*n^2)*x^3)*P(n-1,x) - (25+135*x-(45*n^2-45*n-207)*x^2-(162*n^2-162*n-81)*x^3-(81*n^2-81*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(90*n + (45*n^2 + 171*n)*x - (81*n^3 - 162*n^2 - 81*n)*x^2 - (81*n^3 - 81*n^2)*x^3)*P[n - 1, x] - (25 + 135*x - (45*n^2 - 45*n - 207)*x^2 - (162*n^2 - 162*n - 81)*x^3 - (81*n^2 - 81*n)*x^4)*D[P[n - 1, x], x])/(-60*n + 60 - (54*n - 54)*x + (54*n^3 - 162*n^2 + 162*n - 54)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(90*n + (45*n^2 + 171*n)*x - (81*n^3 - 162*n^2 - 81*n)*x^2 - (81*n^3 - 81*n^2)*x^3)*v[i - 1] - (25 + 135*x - (45*n^2 - 45*n - 207)*x^2 - (162*n^2 - 162*n - 81)*x^3 - (81*n^2 - 81*n)*x^4)*deriv(v[i - 1], x))/(-60*n + 60 - (54*n - 54)*x + (54*n^3 - 162*n^2 + 162*n - 54)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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