Real-rooted atlas

A280580 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-n + 2\,n^{2}\right)\,P_{n}(x) = \left(12 + 6\,x - 28\,n - 12\,n x + 14\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 6\,x + 3\,x^{2} + 12\,n + 8\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
398118
45163181133
53235082303211094
62188417525906105019614
71551134788274059260133572188372
8113634292845279664123977264252569981839213
985346724845762787876153451243925064456450812818168774
106536382212110112734236018007164651823213067461400287380162181069466
115085201918199284026497993520335104089365950229778643407670278280114452001817217244
1240076322315680231492544698310223101301511560487403651181387022030480160305152951699524218344433442
13319272779713557124980242633811422393284185614306004405537978672012829841641916148961735578990090024354288186124747292
142566981847611756423251323001807537825216309718617096468389074779061357214065611763995210076476836932351000651501786338783381897074213052
152080232782091022108819228217027064117526181380035721986448434971992427363928332756985561750538588801126588362910991900926704597924049484595539219417661956520
Technical details
Node id
n-38f05709377aafdf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-886b9c97c4b2ddcc

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:       2,        1;
   3:       9,        8,        1;
   4:      51,       63,       18,        1;
   5:     323,      508,      230,       32,       1;
   6:    2188,     4175,     2590,      610,      50,       1;
   7:   15511,    34788,    27405,     9260,    1335,      72,      1;
   8:  113634,   292845,   279664,   123977,   26425,    2569,     98,    1;
   9:  853467,  2484576,  2787876,  1534512,  439250,   64456,   4508,  128,   1;
  10: 6536382, 21211011, 27342360, 18007164, 6518232, 1306746, 140028, 7380, 162, 1;
%F Formula
The row polynomials satisfy (2*n^2-n)*P(n,x) = (14*n^2-28*n+12+(6*n^2-12*n+6)*x)*P(n-1,x) + (12*n-9+(8*n-6)*x-(4*n-3)*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[((14*n^2 - 28*n + 12 + (6*n^2 - 12*n + 6)*x)*P[n - 1, x] + (12*n - 9 + (8*n - 6)*x - (4*n - 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 - n)]];
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] = ((14*n^2 - 28*n + 12 + (6*n^2 - 12*n + 6)*x)*v[i - 1] + (12*n - 9 + (8*n - 6)*x - (4*n - 3)*x^2)*deriv(v[i - 1], x))/(2*n^2 - n)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.