Real-rooted atlas

A143495 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2516
32611138
414194181254
57997412232611790
647365691238043535113278
72937143642236566000755451103470
819049733853522723474676130201211561845230
91291020267617521532018767221979112546618346817222398
10913127521654472203787489958621278525146357546116265881164420798
1167310847179815603194181571111188720373437557697697989457786603720951598610126
12516369838153456760718726527601233066351485771000120257467191323441963137127875506011015783913550
134116416797134690810001834233598013680267568620629268618031397343444552104371298836710521998156721126158009579678
14340511649851216155747571829062355641526096472687855260041226304421286589813796089837948293234999653567230201687491431602936243294
1529187139968211295570846551859788165509171799381535699110515740637698564103297472595220174245717362169255517187754567111621514437161119316116484567057902
Technical details
Node id
n-4cadf5ffbd23b8ea · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b056b97fb4477666

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:      26,       11,        1;
   4:     141,       94,       18,       1;
   5:     799,      741,      223,      26,       1;
   6:    4736,     5691,     2380,     435,      35,      1;
   7:   29371,    43642,    23656,    6000,     755,     45,     1;
   8:  190497,   338535,   227234,   74676,   13020,   1211,    56,    1;
   9: 1291020,  2676175,  2153201,  876722,  197911,  25466,  1834,   68,  1;
  10: 9131275, 21654472, 20378748, 9958621, 2785251, 463575, 46116, 2658, 81, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (1+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] + (1 + 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] + (1 + 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

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.