Real-rooted atlas

A063007 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - x + 2\,x^{2} + 12\,n + n x - 2\,n x^{2} - 8\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-2 + x + 10\,x^{2} + 8\,x^{3} + 8\,n + 7\,n x - 10\,n x^{2} - 8\,n x^{3} - 12\,n^{2} - 16\,n^{2} x + 6\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(2 - 18\,x^{2} - 24\,x^{3} - 8\,x^{4} - 8\,n - 24\,n x - 16\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
27411
3498424157
43279848401602311
521219240138007840112034121
61350377340167160165760705608064502387
784841603372171528024595201824480620928591367367557
85278874489072158548322989728032044320189826565381376439296107616719
93259921322765921363874403195561604502836803851527681898856964612608032947201566223057
10200113832261637001113258960312134592054644990406142840704437044608018450432003920716802489344022720574107
11122246433155314170087306080402852211648059755856160834455381767799920128047496683520175312051203310827520189190144328656614573
127437815911049486407266345210360247578371040604428945120101004122419211726079805449378354585604991072486401636494745602781095116814447247364742088234583
134510043097699810480724914943553522063108567520575405839008011204938128384155652979722241545891585638410802638978560510529603584015057643560962326006824961107622297668269680637081
1427268265391461499173772356539761396016630723549440521821431331201160912090257921879215767869442232992350617601937355554534401201929714585605108077130956813690211598336193833902080085201715200980902103166083
151644551262333014953710300254124497390401304174226700804547835942249601138339229803776210520706043648029148203862835203025342312312320232993196147712012995898561085445017529008128001232229806080001610312417280065727037440014068759957860373
Technical details
Node id
n-7529eeb526a60491 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ef46ce37242ffd3c

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:       49,        84,         24;
   4:      327,       984,        840,        160;
   5:     2121,      9240,      13800,       7840,       1120;
   6:    13503,     77340,     167160,     165760,      70560,       8064;
   7:    84841,    603372,    1715280,    2459520,    1824480,     620928,      59136;
   8:   527887,   4489072,   15854832,   29897280,   32044320,   18982656,    5381376,     439296;
   9:  3259921,  32276592,  136387440,  319556160,  450283680,  385152768,  189885696,   46126080,   3294720;
  10: 20011383, 226163700, 1113258960, 3121345920, 5464499040, 6142840704, 4370446080, 1845043200, 392071680, 24893440;
%F Formula
The row polynomials satisfy (2*n^3-8*n^2+12*n-6+(n-1)*x-(2*n-2)*x^2)*P(n,x) = (6*n^3-12*n^2+8*n-2+(8*n^3-16*n^2+7*n+1)*x-(10*n-10)*x^2-(8*n-8)*x^3)*P(n-1,x) + (4*n^2-8*n+2+(12*n^2-24*n)*x+(8*n^2-16*n-18)*x^2-24*x^3-8*x^4)*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[((6*n^3 - 12*n^2 + 8*n - 2 + (8*n^3 - 16*n^2 + 7*n + 1)*x - (10*n - 10)*x^2 - (8*n - 8)*x^3)*P[n - 1, x] + (4*n^2 - 8*n + 2 + (12*n^2 - 24*n)*x + (8*n^2 - 16*n - 18)*x^2 - 24*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 8*n^2 + 12*n - 6 + (n - 1)*x - (2*n - 2)*x^2)]];
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] = ((6*n^3 - 12*n^2 + 8*n - 2 + (8*n^3 - 16*n^2 + 7*n + 1)*x - (10*n - 10)*x^2 - (8*n - 8)*x^3)*v[i - 1] + (4*n^2 - 8*n + 2 + (12*n^2 - 24*n)*x + (8*n^2 - 16*n - 18)*x^2 - 24*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 8*n^2 + 12*n - 6 + (n - 1)*x - (2*n - 2)*x^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

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