Real-rooted atlas

A143495 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
31725951
47716711827389
537210611108499813121
61915675493016214199624326423
710481439967514166464318517741729236403
8608142960156034226644394237811548812950621872235045
937293920654314894772645444751480112520531729590111463656122303745
102409837149670954044914562187586596249653669959914342945337253641960819683234492999
111636078611265779934211759260130469267431510450104084724763238679328270154267091580017590492591823251
1211639320588034862229691554325876470632755373330965872050083948293658161001303043089604970222987596577417714730048874853
13865549453713572609226475829942583031676368458128507584783596182598596348362974919390210207188278231504959331933744822623067531441364618712945
146713065156599317471992427006216845888923253069523703113221079937903901879030143850516577595104217158104879636268442801236409402914545715468624074015943234620845163319
155419036045352098400711322874073139596066143036654108272435113711371382820007112658424496433860276938478042937148622951549997054462392295340867658768175646648446351330614053478296961039858229395
Technical details
Node id
n-f623710345608480 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b373e0a412d5b427

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:      17,       25,        9;
   4:      77,      167,      118,       27;
   5:     372,     1061,     1108,      499,       81;
   6:    1915,     6754,     9301,     6214,     1996,      243;
   7:   10481,    43996,    75141,    66464,    31851,     7741,      729;
   8:   60814,   296015,   603422,   664439,   423781,   154881,    29506,    2187;
   9:  372939,  2065431,  4894772,  6454447,  5148011,  2520531,   729590,  111463,   6561;
  10: 2409837, 14967095, 40449145, 62187586, 59624965, 36699599, 14342945, 3372536, 419608, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) - (1+2*x+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[(n + 2 + (n + 1)*x)*P[n - 1, x] - (1 + 2*x + x^2)*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] = (n + 2 + (n + 1)*x)*v[i - 1] - (1 + 2*x + x^2)*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

Same fingerprint ?

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.