Real-rooted atlas

A033282 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - n + 2\,n x^{2} + 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-6 - 12\,x + 8\,x^{3} - 3\,n - 10\,n x + 6\,n x^{2} + 20\,n x^{3} - 2\,n^{2} x + 9\,n^{2} x^{2} + 16\,n^{2} x^{3} + 3\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 3\,x + 6\,x^{3} + 4\,x^{4} + 4\,n x^{2} + 12\,n x^{3} + 8\,n x^{4} + 2\,n^{2} x^{2} + 6\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2116060131
3453727564481621
4197218467208064336020525
59031256051840940807920025344263927
642797160437098090288011484007413121921923431647
7207934067002536380774928013213200128768646726720146432044994257
8103049230683216824192618858241324523201729728001363281925974425611202048593819513
9518859130817521092424324700317441211202720198509875220958013441384980480522422784859955207878376387
10264672374213100698263020344084832010372252800204661416962711235072023992750080136410393604514764800662165504104977436123
1113648869421298724440993322024490139520845593056001951853460483112687226883455630745602630197785601311597056003866027827251121029121403863334573
12710393732393631000275899219201704930163206635086128001754349290496327275300352043733484748804175162703360278655252480012371289047043286351872003955793920018831544249573
133726935191361153539217130884659211659334269445049736211520150500523064323216038528102450269713162240577595854218244834375749632028719063859200114876255436802776967331840306726174720253274839291247
1419688015197747195453210571004891727857222950304374851115839201243419556579202995113422421125362599834408967206877740142087258906566912005406962085324802893386346905601052933446656002334953005056023826051072003414230910872183
Technical details
Node id
n-ea17f216ed68e6b8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d2d860f7089d1e85

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:      3,        8;
  2:     11,       60,        60;
  3:     45,      372,       756,       448;
  4:    197,     2184,      6720,      8064,       3360;
  5:    903,    12560,     51840,     94080,      79200,      25344;
  6:   4279,    71604,    370980,    902880,    1148400,     741312,     192192;
  7:  20793,   406700,   2536380,   7749280,   13213200,   12876864,    6726720,    1464320;
  8: 103049,  2306832,  16824192,  61885824,  132452320,  172972800,  136328192,   59744256,  11202048;
  9: 518859, 13081752, 109242432, 470031744, 1211202720, 1985098752, 2095801344, 1384980480, 522422784, 85995520;
%F Formula
The row polynomials satisfy -(n+2-(n^3+3*n^2+2*n)*x^2)*P(n,x) = -(3*n+6+(2*n^2+10*n+12)*x-(3*n^3+9*n^2+6*n)*x^2-(4*n^3+16*n^2+20*n+8)*x^3)*P(n-1,x) - (1+3*x-(2*n^2+4*n)*x^2-(6*n^2+12*n+6)*x^3-(4*n^2+8*n+4)*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[(-(3*n + 6 + (2*n^2 + 10*n + 12)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (4*n^3 + 16*n^2 + 20*n + 8)*x^3)*P[n - 1, x] - (1 + 3*x - (2*n^2 + 4*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (4*n^2 + 8*n + 4)*x^4)*D[P[n - 1, x], x])/(-n - 2 + (n^3 + 3*n^2 + 2*n)*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] = (-(3*n + 6 + (2*n^2 + 10*n + 12)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (4*n^3 + 16*n^2 + 20*n + 8)*x^3)*v[i - 1] - (1 + 3*x - (2*n^2 + 4*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (4*n^2 + 8*n + 4)*x^4)*deriv(v[i - 1], x))/(-n - 2 + (n^3 + 3*n^2 + 2*n)*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