Real-rooted atlas

A166960 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + n + n x\right)\,P_{n-1}(x) + \left(2 + x - x^{2}\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
288117
35072241147
44428304506211785
552121199289482312146128611
67836421266420175677476103443241580929
7145575245414285215752266336057439041796692114493171
8326161281147664721544279329845139229787110376849015662614421433975593
98645500883382108152520969320039801292921558296492292170336224201285550842958115309925731
102669365367211478912005619892625652817698902590485216030968215054737922569161576123364248188726460081626813980017
119477545972324439232958768853910614289686641283195524954309611216158237567896826454798148820613029376637862442622073612128129412712414803
12382774879673441938199356500164094023920622084661873055973763087073569942721196779545914402609527802128829473899958081531894056663138624662200329142439011565271448967385
131742329486967264947724934840763221790942069603968274979288570694402068984983601256094484568393712802559610364102400388186740785832301433737743001068125636920148441533566339168048938193625779949325571
14886713251405321605153437120913121921280473058311175248177248592894474169614930737152357892007854325612633545442553065696519362564925609121208108852730060406536042863761075461207063053066916680119762041979565209806016245609475262219809
Technical details
Node id
n-fe127344ae43d1f4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-abddcc76b66eba40

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         2,          1;
  2:         8,          8,          1;
  3:        50,         72,         24,          1;
  4:       442,        830,        450,         62,          1;
  5:      5212,      11992,       8948,       2312,        146,         1;
  6:     78364,     212664,     201756,      77476,      10344,       324,        1;
  7:   1455752,    4541428,    5215752,    2663360,     574390,     41796,      692,      1;
  8:  32616128,  114766472,  154427932,   98451392,   29787110,   3768490,   156626,   1442,    1;
  9: 864550088, 3382108152, 5209693200, 3980129292, 1558296492, 292170336, 22420128, 555084, 2958, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+1+n*x)*P(n-1,x) + (2+x-x^2)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 1 + n*x)*P[n - 1, x] + (2 + x - x^2)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 1 + n*x)*v[i - 1] + (2 + x - x^2)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*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 - 1, v[i])))};
T(9)

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.