Real-rooted atlas

A059110 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(4 + 2\,x\right)\,P_{n-1}'(x) + \left(2 + x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
386115
4444212199
5304344132201801
62512324015203203017623
7240643451218840496066042183079
8261536410032252672786801330012185611017495
9317388853725443661504130323226376031024207272113808097
10424839687691788857129984227136005306112753984651843312901205374123
116211594241194018560955889280417569280109845120181520641908480126240504011013318673599
1298419502081996695782417081884160809818944023566118404413319685427206443876802290207370132157845821707
131678792683523577068963843247843568641655058099205257391040010951852032153392870414568998493297603938001042815611081091446785
14306572354969668326201415686547291465728355897440870412213547174402790662434564367008588847426945283580331041859572064750414352182121553820597871
155965109352857613857681167769613949042151116880379668561920295553764746247326720617216126253577049615373789516813312178880817280464350990641025024192922101456410531639799
Technical details
Node id
n-b27c092771eb3329 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cc63c8292327778d

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:        8,        6,        1;
   4:       44,       42,       12,        1;
   5:      304,      344,      132,       20,       1;
   6:     2512,     3240,     1520,      320,      30,      1;
   7:    24064,    34512,    18840,     4960,     660,     42,     1;
   8:   261536,   410032,   252672,    78680,   13300,   1218,    56,    1;
   9:  3173888,  5372544,  3661504,  1303232,  263760,  31024,  2072,   72,  1;
  10: 42483968, 76917888, 57129984, 22713600, 5306112, 753984, 65184, 3312, 90, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (4+2*x)*d/dx P(n-1,x) + (2+x)*d^2/dx^2 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] + (4 + 2*x)*D[P[n - 1, x], x] + (2 + x)*D[P[n - 1, x], {x, 2}]];
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] + (4 + 2*x)*deriv(v[i - 1], x) + (2 + x)*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.