Real-rooted atlas

A166973 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
399119
46382211167
55368473493811771
6530799555605101560121943
76038913104292440240602360871310379
87759881902083160122254866078155473911914910967
91106239130102789293466471249154223865452096788582156185608331
101726387275152483845696356832894337617023748283053114897621439419811626003703
1129215193749475734245116923829956894759260204229114030917373924481485103026022755245133361395499
125322170997318617548903925320571908516974911321559554507815111244266431128640359635113051996995343202971734225149047
1310373208651413888507324096577088081628443300308000401758058951220393773237946494638138063558438444148929495362488549819354117231746851531
1421517178350762859656802966531380895391201041145425834413645285165867553013873850789942210116062527818708321603499765970593217900406234943700574161429139039208183
154728627581847892004172331628373346117890135836131423466311338921624125344051574490480448762304876589033600289422303500896620580844312254124760896498602111025315295914483111292894275510379
Technical details
Node id
n-7fcf0d93134e505b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0f86ba650a12e57e

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,         9,         1;
   4:        63,        82,        21,         1;
   5:       536,       847,       349,        38,        1;
   6:      5307,      9955,      5605,      1015,       60,       1;
   7:     60389,    131042,     92440,     24060,     2360,      87,      1;
   8:    775988,   1902083,   1601222,    548660,    78155,    4739,    119,     1;
   9:  11062391,  30102789,  29346647,  12491542,  2386545,  209678,   8582,   156,   1;
  10: 172638727, 515248384, 569635683, 289433761, 70237482, 8305311, 489762, 14394, 198, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (5+5*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[(2 + x)*P[n - 1, x] + (5 + 5*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] = (2 + x)*v[i - 1] + (5 + 5*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.