Real-rooted atlas

A197208 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(30 + 91\,n + 102\,n^{2} + 52\,n^{3} + 12\,n^{4} + n^{5}\right)\,P_{n}(x) = \left(120 + 240\,x + 274\,n + 706\,n x + 225\,n^{2} + 789\,n^{2} x + 85\,n^{3} + 418\,n^{3} x + 15\,n^{4} + 105\,n^{4} x + n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(303\,x - 195\,x^{2} + 450\,n x - 465\,n x^{2} + 225\,n^{2} x - 384\,n^{2} x^{2} + 45\,n^{3} x - 129\,n^{3} x^{2} + 3\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(83\,x^{2} + 83\,x^{3} + 111\,n x^{2} + 111\,n x^{3} + 46\,n^{2} x^{2} + 46\,n^{2} x^{3} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 30 + 240\,x + 90\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
13024090360
260120018002403300
3105420015750840052528980
41681176088200117600294001008248136
5252282243704409878406174008467217642090592
6360604801270080592704074088002540160211680288017421480
7495118800374220027941760611226004191264087318004752004455144049950
866021780098010001097712003841992004610390401920996002613600098010066001184251200

All 15 rows

Technical details
Node id
n-f1b29403ec598a4b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f7f5c9aa0a83f4e6

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:   30,    240,       90;
   2:   60,   1200,     1800,        240;
   3:  105,   4200,    15750,       8400,        525;
   4:  168,  11760,    88200,     117600,      29400,        1008;
   5:  252,  28224,   370440,     987840,     617400,       84672,        1764;
   6:  360,  60480,  1270080,    5927040,    7408800,     2540160,      211680,        2880;
   7:  495, 118800,  3742200,   27941760,   61122600,    41912640,     8731800,      475200,       4455;
   8:  660, 217800,  9801000,  109771200,  384199200,   461039040,   192099600,    26136000,     980100,      6600;
   9:  858, 377520, 23359050,  373744800, 1962160200,  3767347584,  2747024280,   747489600,   70077150,   1887600,    9438;
  10: 1092, 624624, 51531480, 1133692560, 8502694200, 24487759296, 28569052512, 13604310720, 2550808260, 171771600, 3435432, 13104;
%F Formula
The row polynomials satisfy (n^5+12*n^4+52*n^3+102*n^2+91*n+30)*P(n,x) = (n^5+15*n^4+85*n^3+225*n^2+274*n+120+(10*n^5+105*n^4+418*n^3+789*n^2+706*n+240)*x)*P(n-1,x) + ((3*n^4+45*n^3+225*n^2+450*n+303)*x-(15*n^4+129*n^3+384*n^2+465*n+195)*x^2)*d/dx P(n-1,x) + ((6*n^3+46*n^2+111*n+83)*x^2+(6*n^3+46*n^2+111*n+83)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 30 + 240*x + 90*x^2.
%t Mathematica
P[1, x_] := 30 + 240*x + 90*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120 + (10*n^5 + 105*n^4 + 418*n^3 + 789*n^2 + 706*n + 240)*x)*P[n - 1, x] + ((3*n^4 + 45*n^3 + 225*n^2 + 450*n + 303)*x - (15*n^4 + 129*n^3 + 384*n^2 + 465*n + 195)*x^2)*D[P[n - 1, x], x] + ((6*n^3 + 46*n^2 + 111*n + 83)*x^2 + (6*n^3 + 46*n^2 + 111*n + 83)*x^3)*D[P[n - 1, x], {x, 2}])/(n^5 + 12*n^4 + 52*n^3 + 102*n^2 + 91*n + 30)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 30 + 240*x + 90*x^2; for(n = 2, N, my(i = n); v[i] = ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120 + (10*n^5 + 105*n^4 + 418*n^3 + 789*n^2 + 706*n + 240)*x)*v[i - 1] + ((3*n^4 + 45*n^3 + 225*n^2 + 450*n + 303)*x - (15*n^4 + 129*n^3 + 384*n^2 + 465*n + 195)*x^2)*deriv(v[i - 1], x) + ((6*n^3 + 46*n^2 + 111*n + 83)*x^2 + (6*n^3 + 46*n^2 + 111*n + 83)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^5 + 12*n^4 + 52*n^3 + 102*n^2 + 91*n + 30)); 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

Same fingerprint ?

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