Real-rooted atlas

A104684 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4 + x + 2\,x^{2} + 4\,n + n x - n x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 6\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
316321361
46424022863595
52561536249615363216145
610248960217602208010020168365527
7409649152165120243200178320640328989712909
8163842580481139712226688023721601358448403396486397863667
965536131072073400321880883226199040211742729936640251494426572987615745
1026214464880644482662414313062425366118427024076817755852870541568155546281462563983726695
111048576314572802624716801018429440222437376029892280322570695680142138368048942504095600160809745311112210781
124194304149946368148504576068665344001804763136029660725248319722332162301421056010977384000333458136058458589245046719126142119187
1316777216704643072816630988844291850240137514516480270085128192354006073344317843767296196533469440823908940802238719788835597122562515959691437751935361
14671088643271557120438388654082753015316489947067187202293251637248357340938240038839108239362983918519296161606620032060414984115214847899040021599884644140993361916443380994775
15268435456150323855362305357250561658025148416688658684313618370529329152334408421867524303352050483239880227459072267516995665921288352998732843451498388489747150110881306729462087923848253188609259215725
Technical details
Node id
n-3bf3b2c1960395d5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e88a2f797c4f541a

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:      4,       3;
   3:     16,      32,       13;
   4:     64,     240,      228,        63;
   5:    256,    1536,     2496,      1536,       321;
   6:   1024,    8960,    21760,     22080,     10020,      1683;
   7:   4096,   49152,   165120,    243200,    178320,     64032,      8989;
   8:  16384,  258048,  1139712,   2266880,   2372160,   1358448,    403396,    48639;
   9:  65536, 1310720,  7340032,  18808832,  26199040,  21174272,   9936640,  2514944,   265729;
  10: 262144, 6488064, 44826624, 143130624, 253661184, 270240768, 177558528, 70541568, 15554628, 1462563;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(n+1)*x-(n-2)*x^2)*P(n-1,x) + (8*x+6*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (n + 1)*x - (n - 2)*x^2)*P[n - 1, x] + (8*x + 6*x^2 + x^3)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 4 + (n + 1)*x - (n - 2)*x^2)*v[i - 1] + (8*x + 6*x^2 + x^3)*deriv(v[i - 1], x))/(n - 1)); 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