Real-rooted atlas

A039755 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(4 + 2\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3138122
47161151148
54574961782411156
633554361203041035110192
7275094140023491628081548199544
8248127422437281197930511620514636311062544
9243412946098883507428137793630056636736243680112269200
10257419395354299345697116207179085418882836598755163828991152034880
1129139778965903383262203770131886040097057570181070402076522143760574512012008720480
12351032869585628685578839622539504470242117486344703822856185328015047042822572358305143128146692416
13447824603131170416517921309766553828222009584831925393951799707110413132077641414844649890727437360116381681416408360512
14602513988107167780396232920205193896421381763821230593340851389167404304639316567525564034946916345153237195408077124261588619516479403439360
15851875781363725155688916888324026055445832394984740293611256599279289352935887304075523410662711129575371211296327471784007224366255891118936212032241105691505791360
Technical details
Node id
n-7d775ea7c710f376 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-19c065e2707b31bc

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:        3,        1;
   3:       13,        8,        1;
   4:       71,       61,       15,        1;
   5:      457,      496,      178,       24,       1;
   6:     3355,     4361,     2030,      410,      35,      1;
   7:    27509,    41400,    23491,     6280,     815,     48,     1;
   8:   248127,   422437,   281197,    93051,   16205,   1463,    63,    1;
   9:  2434129,  4609888,  3507428,  1377936,  300566,  36736,  2436,   80,  1;
  10: 25741939, 53542993, 45697116, 20717908, 5418882, 836598, 75516, 3828, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (4+2*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[(3 + x)*P[n - 1, x] + (4 + 2*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] = (3 + x)*v[i - 1] + (4 + 2*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.