Real-rooted atlas

A110441 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-96\,n - 72\,n x + 54\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-192 - 360\,x - 216\,x^{2} - 192\,n - 432\,n x - 216\,n x^{2} + 81\,n x^{3} - 72\,n^{2} x + 108\,n^{2} x^{2} + 162\,n^{2} x^{3} + 108\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-64 - 288\,x - 360\,x^{2} - 108\,x^{3} + 72\,n x^{2} + 216\,n x^{3} + 81\,n x^{4} + 72\,n^{2} x^{2} + 216\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
214482789
348264324108744
41641248243017284055975
5560544814688164168100145846670
61912226567840812096093150349925103357181
7652891200386208764640810000472392142884174962691348
822288358656179841643614725921100473558422147025598725904920031139
976096138595280352002311027238419920393135122514758497977602125764196830147608890
102598085283072347807521158589442285010002860945922337582241242915844145239878732006495391078803113
118870401991923214682643255633824012716352001888354944188606880012790275845810421601692738002857971621257647830078912
123028544744330246073807682580940800671580720011559117312136860010561133404876865579819402598156000671623326102036672690873356497464143
131034009627606489624708188161164060979233998486400666366652809142877721689701572096633222580323191401620011203248672260193513635925411622320522405586367270
143530329610174341129910805760512838328321662146179203658229775365715714352326496082841605429158740723335339462401487860014964686884467298794881901249949232717445352898770539285
Technical details
Node id
n-b6a868c0a86892f1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c5030302290ceb8c

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,       6;
  2:    14,      48,      27;
  3:    48,     264,     324,      108;
  4:   164,    1248,    2430,     1728,      405;
  5:   560,    5448,   14688,    16416,     8100,     1458;
  6:  1912,   22656,   78408,   120960,    93150,    34992,     5103;
  7:  6528,   91200,  386208,   764640,   810000,   472392,   142884,   17496;
  8: 22288,  358656, 1798416,  4361472,  5921100,  4735584,  2214702,  559872,   59049;
  9: 76096, 1385952, 8035200, 23110272, 38419920, 39313512, 25147584, 9797760, 2125764, 196830;
%F Formula
The row polynomials satisfy -(96*n+72*n*x-54*n^3*x^2)*P(n,x) = -(192*n+192+(72*n^2+432*n+360)*x-(108*n^3+108*n^2-216*n-216)*x^2-(81*n^3+162*n^2+81*n)*x^3)*P(n-1,x) - (64+288*x-(72*n^2+72*n-360)*x^2-(216*n^2+216*n-108)*x^3-(81*n^2+81*n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n + 192 + (72*n^2 + 432*n + 360)*x - (108*n^3 + 108*n^2 - 216*n - 216)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*P[n - 1, x] - (64 + 288*x - (72*n^2 + 72*n - 360)*x^2 - (216*n^2 + 216*n - 108)*x^3 - (81*n^2 + 81*n)*x^4)*D[P[n - 1, x], x])/(-96*n - 72*n*x + 54*n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(192*n + 192 + (72*n^2 + 432*n + 360)*x - (108*n^3 + 108*n^2 - 216*n - 216)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*v[i - 1] - (64 + 288*x - (72*n^2 + 72*n - 360)*x^2 - (216*n^2 + 216*n - 108)*x^3 - (81*n^2 + 81*n)*x^4)*deriv(v[i - 1], x))/(-96*n - 72*n*x + 54*n^3*x^2)); 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