Real-rooted atlas

A039756 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
266113
3243412171
4116208112201457
5648139610002803013355
64088102329076348059042127509
72864081320855684163698001106561248127
82199206961928445444970561506962374419047212434129
91832224638286487457926026560224145645847251408307290125741939
1016430176623468489501697674889920330258408360016122119210200047101101291397789
111575540486458854721081788224958343056488748480148000336268847042928552188760693013213510328695
12160687904070682488321288667315212658520512732583121625873689605653859207676169664509723300009856156144782460313
1317350255744814265669761603394108161727614869761117905222724515888977611577435584189678860819881919213248092550264136241821602513988107
14197553645440984414166912208003195372824367554848001742255607456792626426592233798028464451071603845713999720474786312256536288816081838221018518757813637
Technical details
Node id
n-a65116365f28afe4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f9cef97b53c8c582

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:       6,       6,       1;
  3:      24,      34,      12,       1;
  4:     116,     208,     112,      20,       1;
  5:     648,    1396,    1000,     280,      30,      1;
  6:    4088,   10232,    9076,    3480,     590,     42,     1;
  7:   28640,   81320,   85568,   41636,    9800,   1106,    56,    1;
  8:  219920,  696192,  844544,  497056,  150696,  23744,  1904,   72,  1;
  9: 1832224, 6382864, 8745792, 6026560, 2241456, 458472, 51408, 3072, 90, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (2+2*x)*d/dx 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[(2 + x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];
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] = (2 + x)*v[i - 1] + (2 + 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 - 1, v[i])))};
T(9)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.