Real-rooted atlas

A128813 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 2\,x\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + n + 3\,n x + 2\,n x^{2} + n^{2} + 3\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
28262155
3562864852731100
461643221134113188573335200
59856882483154385625795005561777231654400
62168322365264107343002594057235203129254338147642089107536000
762881288095208044611474813641316882499503345274415895916713479794355990739248096000
823266073634542603042241574751683039249060192063236989283997625668262163479230138125568168317987323291017290560000
9107023938561800681009601345462073400585963848071616391480565954305416453467273790330398036830210652745658140321683904222893684641939139368806720000
10599334055936112717793717769533343531696047748604506749615683427830529003529786482237606551270765638730558990632955776964139183884631670171961703128542632119899525522923274590722240000
114015538174771283492064734848078864868226825284467091191667791216858461045852126844508528445595530283881301511555103311284273591032171971061900738582155458665725797752135652250229397843062001427194663689454574654918144448000000
12317227515806924872263126074920704754115000625631072476718047257871194420331517397141612356616295213140083079341361446178537135140212208394047422956707442610532537271953811112193107539798171234141242870726766871954434266085330450742526067069562199244367491098560682089728000000
132918493145423708167228733952819377152826027321292685474565765836485917319850242742893627018932622504939058364456567939107623803507243005163641854452316903035207747742916443051041074208280836467949334798219930097401515682786645617555655182666933280384843988998984239761269579070034881227372988246171925067302104187574675200000000
14309360273414913064968278260043672942202881028115246974735460940878547043229999253925760412405874316862306825488157415242162495679880240045045849167619056975535609817099200447291654506040163735207575815442550619852079746718527789206288931019803547180867951340496519137078560754304650526631326520170691069558959157407190756049954221260179307025897570051994227618913795767027461172038400000000
Technical details
Node id
n-6c1febe9f417e297 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2b5bdcd14168c4b1

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:           8,           26,            21;
  3:          56,          286,           485,           273;
  4:         616,         4322,         11341,         13188,           5733;
  5:        9856,        88248,        315438,        562579,         500556,         177723;
  6:      216832,      2365264,      10734300,      25940572,       35203129,       25433814,        7642089;
  7:     6288128,     80952080,     446114748,    1364131688,     2499503345,     2744158959,     1671347979,      435599073;
  8:   232660736,   3454260304,   22415747516,   83039249060,   192063236989,   283997625668,   262163479230,   138125568168,    31798732329;
  9: 10702393856, 180068100960, 1345462073400, 5859638480716, 16391480565954, 30541645346727, 37903303980368, 30210652745658, 14032168390422, 2893684641939;
%F Formula
The row polynomials satisfy (2+2*x)*P(n,x) = (n^2+n+2+(3*n^2+3*n+4)*x+(2*n^2+2*n+2)*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^2 + n + 2 + (3*n^2 + 3*n + 4)*x + (2*n^2 + 2*n + 2)*x^2)*P[n - 1, x])/(2 + 2*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^2 + n + 2 + (3*n^2 + 3*n + 4)*x + (2*n^2 + 2*n + 2)*x^2)*v[i - 1])/(2 + 2*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.