Real-rooted atlas

A063007 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
222
3181230
413218060372
5900186012602804300
6591016380168007560126047910
737926131880181440123480415805544521850
8239624100245617312401580040803880216216240245597480
9149709673206001521828017435880118918804828824108108010296059376600
1092744105190372012618144017405388014900886081081000273873605250960437580624579210
115707449035974818010013889601615734120166540374011372961605145940801487772002494206018475606526806550
123493674362449032300768089322014193499320171142171201407047241679884604803092815440781517880116396280775975267844431644
13212922356416430650236573291318601193894301601648906459201585880936641083882159365264437464017808630840399627228053542288832449872702162541860
1412929165886108910210500418447149120969679030320150948209412016629769316161329555907680777294357840329808859380990532342801998912115224337404001352078007240695010094
1578265638990714605608080299790481536076514381143201325555755524016460994710160150751806004801030921851960052645455462601982849428560535422888000981608628001095183180056163240074435657752050
Technical details
Node id
n-6a8ebfe1e41f2334 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ca6d936c14642db2

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 -2)
  • 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:       0;
   2:       2;
   3:      18,       12;
   4:     132,      180,        60;
   5:     900,     1860,      1260,       280;
   6:    5910,    16380,     16800,      7560,      1260;
   7:   37926,   131880,    181440,    123480,     41580,     5544;
   8:  239624,  1002456,   1731240,   1580040,    803880,   216216,    24024;
   9: 1497096,  7320600,  15218280,  17435880,  11891880,  4828824,  1081080,  102960;
  10: 9274410, 51903720, 126181440, 174053880, 149008860, 81081000, 27387360, 5250960, 437580;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (3*n+2*n*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 2*n*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((3*n + 2*n*x)*v[i - 1] + (4 + 6*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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