Real-rooted atlas

A033282 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 3\,n + 3\,n^{2}\right)\,P_{n}(x) = \left(27\,n + 54\,n x + 9\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n + 18\,n x + 18\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
24590135
3279113411342547
4163810080204121360845738
5942077760238140320760160380806460
653703556470228541546510204691115187644614114169
730502538045701961536553513460814864056567561021891870246292305
8173012425236288156648492536431896109459350013310256968931882962551966564294050948
998113141638636481189767852490537101612561953040204621318722070572868011901444048297536101274875432482
10556598251047394530870964731042007623840129512302920264708531360358696299780310759927920156206453130347125451401306416385755
1131597404366148998306199066566034246518768012351572679603039038823744516623545746059919193303204537997728110202667859702040533571940422813749651231
121795223250413848828804315604475602687209881840111017416039203195317329776065382413869080951154253359209641199611538064853715104640260572962474004737690226800398775402236430
131020865154425696326988829512689869522045143165665695238612251640313994475935712751542030078840131583555539092816726467986719921505532672072000910847266603560332585853921360554309756535606977324113144632
14581039658991585650733758198885955929571518147019148767868514381477752924246899479906801719149071941116418168351761176251151078439210952834478907815555022941495048617145126105469611849004219683021627255649182003327270122200609189148973
Technical details
Node id
n-e6acc766da1264ad · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c9c0a2736821f4fb

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       6;
  2:      45,        90;
  3:     279,      1134,       1134;
  4:    1638,     10080,      20412,      13608;
  5:    9420,     77760,     238140,     320760,      160380;
  6:   53703,    556470,    2285415,    4651020,     4691115,     1876446;
  7:  305025,   3804570,   19615365,   53513460,    81486405,    65675610,    21891870;
  8: 1730124,  25236288,  156648492,  536431896,  1094593500,  1331025696,   893188296,   255196656;
  9: 9811314, 163863648, 1189767852, 4905371016, 12561953040, 20462131872, 20705728680, 11901444048, 2975361012;
%F Formula
The row polynomials satisfy (3*n^2+3*n-6)*P(n,x) = (9*n^2+27*n+(18*n^2+54*n)*x)*P(n-1,x) + (4*n+18*n*x+18*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 + 27*n + (18*n^2 + 54*n)*x)*P[n - 1, x] + (4*n + 18*n*x + 18*n*x^2)*D[P[n - 1, x], x])/(3*n^2 + 3*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 + 27*n + (18*n^2 + 54*n)*x)*v[i - 1] + (4*n + 18*n*x + 18*n*x^2)*deriv(v[i - 1], x))/(3*n^2 + 3*n - 6)); 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