Real-rooted atlas

A046716 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(66 + 216\,x + 234\,x^{2} + 84\,x^{3} - 58\,n - 198\,n x - 222\,n x^{2} - 82\,n x^{3} + 18\,n^{2} + 60\,n^{2} x + 66\,n^{2} x^{2} + 24\,n^{2} x^{3} - 2\,n^{3} - 6\,n^{3} x - 6\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 102\,x + 162\,x^{2} + 114\,x^{3} + 30\,x^{4} - 23\,n - 100\,n x - 162\,n x^{2} - 116\,n x^{3} - 31\,n x^{4} + 8\,n^{2} + 34\,n^{2} x + 54\,n^{2} x^{2} + 38\,n^{2} x^{3} + 10\,n^{2} x^{4} - n^{3} - 4\,n^{3} x - 6\,n^{3} x^{2} - 4\,n^{3} x^{3} - n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-30 - 56\,x - 24\,x^{2} + 26\,n + 52\,n x + 24\,n x^{2} - 6\,n^{2} - 12\,n^{2} x - 6\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8 - 34\,x - 46\,x^{2} - 20\,x^{3} + 13\,n + 48\,n x + 59\,n x^{2} + 24\,n x^{3} - 6\,n^{2} - 21\,n^{2} x - 24\,n^{2} x^{2} - 9\,n^{2} x^{3} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(10 + 16\,x - 12\,n - 16\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 18\,x - 10\,x^{2} + 5\,n + 12\,n x + 7\,n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)
\(P_{3}(x) = 5 + 10\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2224
3510318
41650334103
5652883211125791
632619223162204447067930
71957146263253231884145202526798052
8137001250503543934727843300451169761665381429609
9109601118747241141707000032665909536248161058050128640923881885
109864101240873851007428106026904128520030952203064262778810616880113113810448545632
119864101141591434675164277166328012024599115052318703498141026105553753924811703422711120930119344470406
121085051121752563650952630126527193128956476030430105466818650242232462111218293447767258196178140602199012073814512213698491707
1313020613452339592392014296322244346471558500894155925187012782787196801201467719907787590958624354516111270104679903360182864993631433775696135320189732499
1416926797486335157520674227639154003083108266635721915590137246030067764206772333705454372222658163174868015153666106284604844258734016079122269102560008079321848005450214143199647069878
15236975164805512945893881838362437717510155549387029092402336923926075717779380237068920135083695821864556934516712598238271902691302049565427620108400262419581262268998443003838547928982256876761974154143097005511064
Technical details
Node id
n-d7eece827bf52df6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-70ffd85dfde7ce13

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:      2,        2;
   3:      5,       10,        3;
   4:     16,       50,       33,         4;
   5:     65,      288,      321,       112,         5;
   6:    326,     1922,     3162,      2044,       470,        6;
   7:   1957,    14626,    32532,     31884,     14520,     2526,        7;
   8:  13700,   125050,   354393,    472784,    330045,   116976,    16653,        8;
   9: 109601,  1187472,  4114170,   7000032,   6659095,  3624816,  1058050,   128640,       9;
  10: 986410, 12408738, 51007428, 106026904, 128520030, 95220306, 42627788, 10616880, 1131138, 10;
%F Formula
The row polynomials satisfy (2*n-4)*P(n,x) = -(2*n^3-18*n^2+58*n-66+(6*n^3-60*n^2+198*n-216)*x+(6*n^3-66*n^2+222*n-234)*x^2+(2*n^3-24*n^2+82*n-84)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(4*n^3-34*n^2+100*n-102)*x+(6*n^3-54*n^2+162*n-162)*x^2+(4*n^3-38*n^2+116*n-114)*x^3+(n^3-10*n^2+31*n-30)*x^4)*d/dx P(n-3,x) - (6*n^2-26*n+30+(12*n^2-52*n+56)*x+(6*n^2-24*n+24)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(3*n^3-21*n^2+48*n-34)*x+(3*n^3-24*n^2+59*n-46)*x^2+(n^3-9*n^2+24*n-20)*x^3)*d/dx P(n-2,x) + (4*n^2-12*n+10+(4*n^2-16*n+16)*x)*P(n-1,x) - (n^2-5*n+8+(2*n^2-12*n+18)*x+(n^2-7*n+10)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 5 + 10*x + 3*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[3, x_] := 5 + 10*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 18*n^2 + 58*n - 66 + (6*n^3 - 60*n^2 + 198*n - 216)*x + (6*n^3 - 66*n^2 + 222*n - 234)*x^2 + (2*n^3 - 24*n^2 + 82*n - 84)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (4*n^3 - 34*n^2 + 100*n - 102)*x + (6*n^3 - 54*n^2 + 162*n - 162)*x^2 + (4*n^3 - 38*n^2 + 116*n - 114)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*D[P[n - 3, x], x] - (6*n^2 - 26*n + 30 + (12*n^2 - 52*n + 56)*x + (6*n^2 - 24*n + 24)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (3*n^3 - 21*n^2 + 48*n - 34)*x + (3*n^3 - 24*n^2 + 59*n - 46)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 12*n + 10 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (2*n^2 - 12*n + 18)*x + (n^2 - 7*n + 10)*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 5 + 10*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(2*n^3 - 18*n^2 + 58*n - 66 + (6*n^3 - 60*n^2 + 198*n - 216)*x + (6*n^3 - 66*n^2 + 222*n - 234)*x^2 + (2*n^3 - 24*n^2 + 82*n - 84)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (4*n^3 - 34*n^2 + 100*n - 102)*x + (6*n^3 - 54*n^2 + 162*n - 162)*x^2 + (4*n^3 - 38*n^2 + 116*n - 114)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*deriv(v[i - 3], x) - (6*n^2 - 26*n + 30 + (12*n^2 - 52*n + 56)*x + (6*n^2 - 24*n + 24)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (3*n^3 - 21*n^2 + 48*n - 34)*x + (3*n^3 - 24*n^2 + 59*n - 46)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 12*n + 10 + (4*n^2 - 16*n + 16)*x)*v[i - 1] - (n^2 - 5*n + 8 + (2*n^2 - 12*n + 18)*x + (n^2 - 7*n + 10)*x^2)*deriv(v[i - 1], x))/(2*n - 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