Real-rooted atlas

A104684 → basis A013611 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-31 - 20\,x + 19\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-15 - 22\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
27411
3616416141
4595900432641991
56145121608640256025629761
66552716042015136067840140801024460251
77129092081088245160014771204723207372840967272861
8786366726665828376881122857120012409600303206437273616384116619591
98761574533849088055820262451040460828080640094044160183500801835008655361889815041
1098372669542654162608040695040861202944057416325122450165760660602880106168320884736026214430869546411
11111122107815343614752011335151016013919560704010908430848056648761344195651993604376494080592773120419430401048576507406003501
12126142119187666244968004157068835000021755935555201960099384320120030278860850642745753614581923840027671592960321519616019608371241943048382404928711
13143775193536182738501251842146133684217633101306022400337113741888002379199669862411871496175616419048482406410281959424001684013056001702467993690596966416777216139054141487041
1416443380994775102405942910180289838121797952492641133048064559671861548800447391102878720257869141770240107938105786368325733837045766926585036800992644956160883320422404143972352671088642314783947556091
1518860925921572512638554328848003875918422339600719806506135808090250278854095368061006738104320527239046670336025547391723110409165141393408002404312038768644490818748416056946904268804503843635201879048192026843545638647630102736061

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 - n\right)\,P_{n}(x) = \left(-50 - 80\,x - 32\,x^{2} + 25\,n + 40\,n x + 16\,n x^{2}\right)\,P_{n-2}(x) + \left(21 + 12\,x - 14\,n - 8\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 4\,x\)
Technical details
Node id
n-669d4405c319a323 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-94ca338fcc0ee6de)
Fingerprint
f6-20efcd8d91464382

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:         7,          4;
   3:        61,         64,         16;
   4:       595,        900,        432,         64;
   5:      6145,      12160,       8640,       2560,        256;
   6:     65527,     160420,     151360,      67840,      14080,       1024;
   7:    712909,    2081088,    2451600,    1477120,     472320,      73728,      4096;
   8:   7863667,   26665828,   37688112,   28571200,   12409600,    3032064,    372736,     16384;
   9:  87615745,  338490880,  558202624,  510404608,  280806400,   94044160,  18350080,   1835008,   65536;
  10: 983726695, 4265416260, 8040695040, 8612029440, 5741632512, 2450165760, 660602880, 106168320, 8847360, 262144;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (19*n-31+(12*n-20)*x)*P(n-1,x) - (15+22*x+8*x^2)*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[((19*n - 31 + (12*n - 20)*x)*P[n - 1, x] - (15 + 22*x + 8*x^2)*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] = ((19*n - 31 + (12*n - 20)*x)*v[i - 1] - (15 + 22*x + 8*x^2)*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