Real-rooted atlas

A008459 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 10\,x - 18\,x^{2} - 14\,x^{3} - 4\,x^{4} + 4\,n + 17\,n x + 26\,n x^{2} + 17\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
26181337
3209013263305
4704209309003212641
52521890560081905910168323525
69248316308706048065940379268989213445
7343236036160776395010577080501228239624486391961825
8128701544408048042378376435897050727603660300149709626572918205345
9486206563703912480135135002980177243513470420604802595186092744101462563170195525
10184756277134018597150735134401895493603330807484039335303337963201798740905707449080974531600472677
1170543211639628868353203866019301141208640234507873634228434243548374830256029774012245161503493674364504671915122515985
1227041564867480839962722819787367606580546830154840308482643134894432978129184298473575401910971062082153251182129223564251595969143457011569
1310400600202811700181719283299053234283664542882097161794730189936558240277162821936301896418824242419757580139482383040544551052501292916588614099336191365435096485
14401166008424486008180071900486748080001982849428565849495051401288652314950215359722864027434991183602651111511048191285952858099930160512035730280404078265638990792384825313033485491077
Technical details
Node id
n-c3e7c36a90208dd3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c277ed2a39f7c92c

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:     2,      3;
  2:     6,     18,      13;
  3:    20,     90,     132,       63;
  4:    70,    420,     930,      900,      321;
  5:   252,   1890,    5600,     8190,     5910,     1683;
  6:   924,   8316,   30870,    60480,    65940,    37926,     8989;
  7:  3432,  36036,  160776,   395010,   577080,   501228,   239624,    48639;
  8: 12870, 154440,  804804,  2378376,  4358970,  5072760,  3660300,  1497096,  265729;
  9: 48620, 656370, 3912480, 13513500, 29801772, 43513470, 42060480, 25951860, 9274410, 1462563;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n-2+(17*n-10)*x+(26*n-18)*x^2+(17*n-14)*x^3+(4*n-4)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*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[((4*n - 2 + (17*n - 10)*x + (26*n - 18)*x^2 + (17*n - 14)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + 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] = ((4*n - 2 + (17*n - 10)*x + (26*n - 18)*x^2 + (17*n - 14)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + 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

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