Real-rooted atlas

A046716 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12 + 6\,n\right)\,P_{n}(x) = \left(198 + 1944\,x + 6318\,x^{2} + 6804\,x^{3} - 174\,n - 1782\,n x - 5994\,n x^{2} - 6642\,n x^{3} + 54\,n^{2} + 540\,n^{2} x + 1782\,n^{2} x^{2} + 1944\,n^{2} x^{3} - 6\,n^{3} - 54\,n^{3} x - 162\,n^{3} x^{2} - 162\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 306\,x + 1458\,x^{2} + 3078\,x^{3} + 2430\,x^{4} - 23\,n - 300\,n x - 1458\,n x^{2} - 3132\,n x^{3} - 2511\,n x^{4} + 8\,n^{2} + 102\,n^{2} x + 486\,n^{2} x^{2} + 1026\,n^{2} x^{3} + 810\,n^{2} x^{4} - n^{3} - 12\,n^{3} x - 54\,n^{3} x^{2} - 108\,n^{3} x^{3} - 81\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-90 - 504\,x - 648\,x^{2} + 78\,n + 468\,n x + 648\,n x^{2} - 18\,n^{2} - 108\,n^{2} x - 162\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8 - 102\,x - 414\,x^{2} - 540\,x^{3} + 13\,n + 144\,n x + 531\,n x^{2} + 648\,n x^{3} - 6\,n^{2} - 63\,n^{2} x - 216\,n^{2} x^{2} - 243\,n^{2} x^{3} + n^{3} + 9\,n^{3} x + 27\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(30 + 144\,x - 36\,n - 144\,n x + 12\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 54\,x - 90\,x^{2} + 5\,n + 36\,n x + 63\,n x^{2} - n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)
\(P_{3}(x) = 5 + 30\,x + 27\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2268
35302762
416150297108571
565864288930244057247
632657662845855188380701458129266
7195743878292788860868117612061381851032994532
8137003751503189537127651682673364528425168121400371749683659901
9109601356241637027530189000864539386695880830288771318450281335680590492702630573
1098641037226214459066852286272640810410122430231385343583107565745223219116560742139641819683098625029932
119864101424774302607647849344908563240199252831905563444950014102808030909511755983353767678615633472188932651906495394004551584602
121085051125257690950857367113857342144818123855846483810132843693199863078746487891947740777025112476210251238582767473082917071294667241052125764178918999899983
1313020613457018777176012866690019871254732079521676266299401470310621728882240875869967812203172246142651068823259802060424702060414537834880107979950088578725398906321931269087338719308217128091
1416926797486100547256202220487523860270224392319916444155162801116926073064667022455962432712762373483858134028634363160994232033233293241190514954466132209494560908680859045349775122730004982105864459738222320522460156294861048118
15236975164805153883768164543452619394575904199833449785484325892908380120751744203893976075246707784760142535091890786015788049144392504130195355565159452415963118444606400927095613838469464601662671821210020399617499261230624095445297806736027174453526139867793155138464
Technical details
Node id
n-a9aff647c13fb176 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-90434d8020f51a22

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,        6;
   3:      5,       30,        27;
   4:     16,      150,       297,        108;
   5:     65,      864,      2889,       3024,         405;
   6:    326,     5766,     28458,      55188,       38070,        1458;
   7:   1957,    43878,    292788,     860868,     1176120,      613818,        5103;
   8:  13700,   375150,   3189537,   12765168,    26733645,    28425168,    12140037,       17496;
   9: 109601,  3562416,  37027530,  189000864,   539386695,   880830288,   771318450,   281335680,      59049;
  10: 986410, 37226214, 459066852, 2862726408, 10410122430, 23138534358, 31075657452, 23219116560, 7421396418, 196830;
%F Formula
The row polynomials satisfy (6*n-12)*P(n,x) = -(6*n^3-54*n^2+174*n-198+(54*n^3-540*n^2+1782*n-1944)*x+(162*n^3-1782*n^2+5994*n-6318)*x^2+(162*n^3-1944*n^2+6642*n-6804)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(12*n^3-102*n^2+300*n-306)*x+(54*n^3-486*n^2+1458*n-1458)*x^2+(108*n^3-1026*n^2+3132*n-3078)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*d/dx P(n-3,x) - (18*n^2-78*n+90+(108*n^2-468*n+504)*x+(162*n^2-648*n+648)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(9*n^3-63*n^2+144*n-102)*x+(27*n^3-216*n^2+531*n-414)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*d/dx P(n-2,x) + (12*n^2-36*n+30+(36*n^2-144*n+144)*x)*P(n-1,x) - (n^2-5*n+8+(6*n^2-36*n+54)*x+(9*n^2-63*n+90)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 6*x, P(3,x) = 5 + 30*x + 27*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[3, x_] := 5 + 30*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^3 - 54*n^2 + 174*n - 198 + (54*n^3 - 540*n^2 + 1782*n - 1944)*x + (162*n^3 - 1782*n^2 + 5994*n - 6318)*x^2 + (162*n^3 - 1944*n^2 + 6642*n - 6804)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*D[P[n - 3, x], x] - (18*n^2 - 78*n + 90 + (108*n^2 - 468*n + 504)*x + (162*n^2 - 648*n + 648)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*D[P[n - 2, x], x] + (12*n^2 - 36*n + 30 + (36*n^2 - 144*n + 144)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];
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 + 6*x; v[3] = 5 + 30*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-(6*n^3 - 54*n^2 + 174*n - 198 + (54*n^3 - 540*n^2 + 1782*n - 1944)*x + (162*n^3 - 1782*n^2 + 5994*n - 6318)*x^2 + (162*n^3 - 1944*n^2 + 6642*n - 6804)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*deriv(v[i - 3], x) - (18*n^2 - 78*n + 90 + (108*n^2 - 468*n + 504)*x + (162*n^2 - 648*n + 648)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*deriv(v[i - 2], x) + (12*n^2 - 36*n + 30 + (36*n^2 - 144*n + 144)*x)*v[i - 1] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*deriv(v[i - 1], x))/(6*n - 12)); 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