Real-rooted atlas

A059364 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + n + 3\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
26191540
324118193105440
4120806202722629456160
5720615621028358693055210395104720
65040524522272245244476801614699411351352094400
7403204952162604572760393613307993139619438130195202702548171200
8362880514238431859820112713148249038849351893368310524786156456540344594251252451200
936288005831856041630349617324680604631938302825067181197912218527464536334331726851065472907536321084800
10399168007177089605804028216277995220768733315058218802809426328096754840328772555856619324521662476864658535137493105751162274713600
11479001600953059392086155644672467086913880168738681473242649995945427696256748885991496031606189366304465065367015884772193287887320531623414322540679614976000
1262270208001358727609601358288228736822602099724033613201438516976296650973462066763275990553213009028309183640502037061032931869676646861593028224709655243301569205079058535806251545825369088000
138717829120020703482150402268459974822415183807613723269268738706470422743717502027765529469544015112100784734848073371377182722832022813933973047370385101462876415400325035238426404755152637337381410021345804667687563378840132608000
14130767436800033583393670400400309094459520293542453475697614793615013950288542035104779180561488988239161071843115317190485483034988531394842161936083925853318423875562795329968416453697709180007322531689175149729493954746669854076202561902833536293752788668965834752000
Technical details
Node id
n-9ce77fe1ca1469bb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-854a5e7b1601ca9e

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,        3;
  2:       6,       19,        15;
  3:      24,      118,       193,        105;
  4:     120,      806,      2027,       2262,        945;
  5:     720,     6156,     21028,      35869,      30552,      10395;
  6:    5040,    52452,    227224,     524447,     680161,     469941,     135135;
  7:   40320,   495216,   2604572,    7603936,   13307993,   13961943,    8130195,    2027025;
  8:  362880,  5142384,  31859820,  112713148,  249038849,  351893368,  310524786,  156456540,   34459425;
  9: 3628800, 58318560, 416303496, 1732468060, 4631938302, 8250671811, 9791221852, 7464536334, 3317268510, 654729075;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(3*n+2)*x+(2*n+1)*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 + (3*n + 2)*x + (2*n + 1)*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 + (3*n + 2)*x + (2*n + 1)*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.