Real-rooted atlas

A143495 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + x\right)\,P_{n-1}(x) + \left(2 + x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2516
32711139
415797181273
59798112292612046
665176761257044535116329
74610757363274216410770451138117
8345261499969287770848611379012325611232940
9272624344961553023525107697822129126782186268111572905
10226235254179727332122698134096773336417511455482162694811113852037
1119671217140189795534711422116609145048551980902955910795478184037659511170562584
1217873567653996556785382824620220642616616933548601518021342205033621218671324955115110112545888331
13169298973954107968228343179850165258891783709822476741247598023242406196849632660393637220630567871261139854909405
1416680885154143612739175349867370365832887309105313905127135939671022677783551912010826358401045189866960382310882882714311618235286222
15170629904121147782680166915900081863677424206860295419770447600546297877275691410186347572249945297625666695342081819741181780645554611284161119399855236195
Technical details
Node id
n-fbde8ee140a93e8b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f60ae7370def9f51

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:        5,        1;
   3:       27,       11,        1;
   4:      157,       97,       18,        1;
   5:      979,      811,      229,       26,       1;
   6:     6517,     6761,     2570,      445,      35,      1;
   7:    46107,    57363,    27421,     6410,     770,     45,     1;
   8:   345261,   499969,   287770,    84861,   13790,   1232,    56,    1;
   9:  2726243,  4496155,  3023525,  1076978,  221291,  26782,  1862,   68,  1;
  10: 22623525, 41797273, 32122698, 13409677, 3336417, 511455, 48216, 2694, 81, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (2+x)*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[(5 + x)*P[n - 1, x] + (2 + x)*D[P[n - 1, x], x]];
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] = (5 + x)*v[i - 1] + (2 + x)*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.