Real-rooted atlas

A137320 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 6\,x + 5\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
26324280
324176424336960
4120109637045496302413440
5720777633184700167310430240215040
6504062352317824855136128190410158243326403870720
740320559296329081610654976205168642350944014851008399168077414400
8362880555782436888192138675392323166464478304192439281792228988224518918401703116800
9362880060658560446691456190318860851731201289307372416110890766088439827328372475353672648576040874803200
10399168007216761605823484416276354465288545215052817997789849626159042892825917424972816756969881663862646400108972864001062744883200
11479001600929878272081428631552424801108992146759295078435269691904006018731523072729553785958461576243814403447466937856115256977920017435658240029756856729600
12622702080012902720256012166515164166906701153280263003272130567079967963852813820198603673619716042806681620407326057164814949668467660873590345073152218603218176002964061900800892705701888000
138717829120019184672102401935561087590411859354344140849252520174182414646014047744003209222038007808524788174159667264059133532057605766272275762176372120515520307216306707167631364349826593280005335311421440028566582460416000
14130767436800030433395686400326785040133120214665975826329696411553515141123132699990471065675965757260830720139693444846098432195798453388423168208206437847342080165377250567527424951629587003054083750748350841958490649672404480001013709170073600971263803654144000
Technical details
Node id
n-4fda2e5d70c9292d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1a2f142e31a877af

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,        6;
  2:       6,       32,        42;
  3:      24,      176,       424,        336;
  4:     120,     1096,      3704,       5496,       3024;
  5:     720,     7776,     33184,      70016,      73104,      30240;
  6:    5040,    62352,    317824,     855136,    1281904,    1015824,      332640;
  7:   40320,   559296,   3290816,   10654976,   20516864,   23509440,    14851008,    3991680;
  8:  362880,  5557824,  36888192,  138675392,  323166464,  478304192,   439281792,  228988224,   51891840;
  9: 3628800, 60658560, 446691456, 1903188608, 5173120128, 9307372416, 11089076608, 8439827328, 3724753536, 726485760;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+6)*x+(n+5)*x^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[Together[((n + 1 + (2*n + 6)*x + (n + 5)*x^2)*P[n - 1, x])/(1 + 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] = ((n + 1 + (2*n + 6)*x + (n + 5)*x^2)*v[i - 1])/(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

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.