Real-rooted atlas

A063007 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-16 - 18\,x + 39\,x^{2} + 27\,x^{3} - 4\,x^{4} + 16\,n + 18\,n x - 99\,n x^{2} - 63\,n x^{3} + 16\,n x^{4} + 80\,n^{2} x^{2} + 48\,n^{2} x^{3} - 25\,n^{2} x^{4} - 20\,n^{3} x^{2} - 12\,n^{3} x^{3} + 19\,n^{3} x^{4} - 7\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-48 - 70\,x - 3\,x^{2} + 33\,x^{3} + 12\,x^{4} + 48\,n + 14\,n x - 189\,n x^{2} - 209\,n x^{3} - 60\,n x^{4} + 56\,n^{2} x + 252\,n^{2} x^{2} + 240\,n^{2} x^{3} + 51\,n^{2} x^{4} - 8\,n^{2} x^{5} - 60\,n^{3} x^{2} - 56\,n^{3} x^{3} + 15\,n^{3} x^{4} + 16\,n^{3} x^{5} - 8\,n^{4} x^{3} - 21\,n^{4} x^{4} - 10\,n^{4} x^{5} + 3\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(16 + 72\,x + 76\,x^{2} - 6\,x^{3} - 38\,x^{4} - 12\,x^{5} + 112\,n x^{2} + 264\,n x^{3} + 200\,n x^{4} + 48\,n x^{5} - 56\,n^{2} x^{2} - 132\,n^{2} x^{3} - 84\,n^{2} x^{4} + 8\,n^{2} x^{6} - 16\,n^{3} x^{4} - 24\,n^{3} x^{5} - 8\,n^{3} x^{6} + 4\,n^{4} x^{4} + 6\,n^{4} x^{5} + 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
23811
3137254139
4635288103201721
5321360083706720175020761
61683236407371089600472509072244955
78989151704593460967680771750299376452762838235
8486399584964511052923328098752505787936176576421964832400065
926572959883843294270081164160108974250856215363943539698841601042470365318785
10146256337097640233566740672967680108783675010728637926621615002503987205316597048620004076383355
1180974532282979601618866810534074112010098338250119909069289287918640470486016015063691502771340002235547645083885947
12450467191397469744110206453504096476384088709370750123222363264114908858064730373529603131475633086835320001408394988101582208494814136217
13251595969851689425673937926062305755369920746183938500118721265062412951360982569909779742725330242932301978736760004835489458870091723524570023645394691486393
141409933619517166635444900959472502231718128640606049393950010868271077664135809782748641215593972736078701053731303664542882000119854413478826167576780834275177300203851760058471805545067
15792384825331306255596032157252363601598882568192047821488214500954400143977281344314567996401378302226329601043808375109505849495051400023992478085576700917235200013824321511001651199256009026235000630482736141547
Technical details
Node id
n-6ec298dee79c9c3c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a9fc593edbc03cee

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,        8;
   3:      13,       72,        54;
   4:      63,      528,       810,       320;
   5:     321,     3600,      8370,      6720,       1750;
   6:    1683,    23640,     73710,     89600,      47250,       9072;
   7:    8989,   151704,    593460,    967680,     771750,     299376,     45276;
   8:   48639,   958496,   4511052,   9233280,    9875250,    5787936,   1765764,    219648;
   9:  265729,  5988384,  32942700,  81164160,  108974250,   85621536,  39435396,   9884160,  1042470;
  10: 1462563, 37097640, 233566740, 672967680, 1087836750, 1072863792, 662161500, 250398720, 53165970, 4862000;
%F Formula
The row polynomials satisfy (16*n-16+(18*n-18)*x-(20*n^3-80*n^2+99*n-39)*x^2-(12*n^3-48*n^2+63*n-27)*x^3+(n^5-7*n^4+19*n^3-25*n^2+16*n-4)*x^4)*P(n,x) = (48*n-48+(56*n^2+14*n-70)*x-(60*n^3-252*n^2+189*n+3)*x^2-(8*n^4+56*n^3-240*n^2+209*n-33)*x^3+(3*n^5-21*n^4+15*n^3+51*n^2-60*n+12)*x^4+(2*n^5-10*n^4+16*n^3-8*n^2)*x^5)*P(n-1,x) + (16+72*x-(56*n^2-112*n-76)*x^2-(132*n^2-264*n+6)*x^3+(4*n^4-16*n^3-84*n^2+200*n-38)*x^4+(6*n^4-24*n^3+48*n-12)*x^5+(2*n^4-8*n^3+8*n^2)*x^6)*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[((48*n - 48 + (56*n^2 + 14*n - 70)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (8*n^4 + 56*n^3 - 240*n^2 + 209*n - 33)*x^3 + (3*n^5 - 21*n^4 + 15*n^3 + 51*n^2 - 60*n + 12)*x^4 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^5)*P[n - 1, x] + (16 + 72*x - (56*n^2 - 112*n - 76)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (4*n^4 - 16*n^3 - 84*n^2 + 200*n - 38)*x^4 + (6*n^4 - 24*n^3 + 48*n - 12)*x^5 + (2*n^4 - 8*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(16*n - 16 + (18*n - 18)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (12*n^3 - 48*n^2 + 63*n - 27)*x^3 + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^4)]];
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] = ((48*n - 48 + (56*n^2 + 14*n - 70)*x - (60*n^3 - 252*n^2 + 189*n + 3)*x^2 - (8*n^4 + 56*n^3 - 240*n^2 + 209*n - 33)*x^3 + (3*n^5 - 21*n^4 + 15*n^3 + 51*n^2 - 60*n + 12)*x^4 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^5)*v[i - 1] + (16 + 72*x - (56*n^2 - 112*n - 76)*x^2 - (132*n^2 - 264*n + 6)*x^3 + (4*n^4 - 16*n^3 - 84*n^2 + 200*n - 38)*x^4 + (6*n^4 - 24*n^3 + 48*n - 12)*x^5 + (2*n^4 - 8*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(16*n - 16 + (18*n - 18)*x - (20*n^3 - 80*n^2 + 99*n - 39)*x^2 - (12*n^3 - 48*n^2 + 63*n - 27)*x^3 + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^4)); 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