Real-rooted atlas

A059364 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + 2\,n + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
2155245112
31055599915852240
494568161842222112994558240
510395948213457786300945737472088451863680
613513514925486865639168356722321106117058660522112570819840
72027025263071351462684774516386118364004039290006695730180151514126253116072960
834459425514113120335469956412504716128291228810144339422467240398328332214836091204996616625155803648000
9654729075110431480058276147715636171349030010160092360021902036866286237315455117219029267215648898292533151848748151258725004288000
1013749310575258750000180219076008848110989203859696361664470583428159915286808812781975709233813726079775089696706409903739403653705035407579867420125540950265856000
11316234143225656996898001562031232043163351335892754653132634245602818635042706335914146611816292187734890888741742581884009833265763175280191967249675199077862332217534109403390562536784618078208000
12790585358062517974469751840018727092810998101182292768898131250374020145682647152597546185166464337004668350672636546701183752844416646560066618273007543652982183481408308498871978288450106075503390427200167136076613756252722061737787392000
13213458046676875527211707276992560089619558170070418472639500785354198671371144944100567899568147206748191718679599328198376432622179383912448940464322845375941252374894631404972246738737143077599138302774192144788063908222506073269086549783475885821206052908125217764939022991360000
146190283353629375165058503770909700204310964033481775515560814860338469256814676380835785543233101514291795177070368854446222442559938391925690573750534093808320600047647323922145340660833260848166927323867089151209191421505267437530853621710336812713500954742205890253718651529088719937005049180874501576312518727784755977256960000
Technical details
Node id
n-e8fd873e4e7b5233 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-23e8f34590336d7d

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:         3,           5;
  2:        15,          52,          45;
  3:       105,         559,         991,          585;
  4:       945,        6816,       18422,        22112,          9945;
  5:     10395,       94821,      345778,       630094,        573747,        208845;
  6:    135135,     1492548,     6865639,     16835672,      23211061,      17058660,       5221125;
  7:   2027025,    26307135,   146268477,    451638611,     836400403,     929000669,     573018015,     151412625;
  8:  34459425,   514113120,  3354699564,  12504716128,   29122881014,   43394224672,   40398328332,   21483609120,   4996616625;
  9: 654729075, 11043148005, 82761477156, 361713490300, 1016009236002, 1902036866286, 2373154551172, 1902926721564, 889829253315, 184874815125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(6*n+2)*x+(4*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[((2*n + 1 + (6*n + 2)*x + (4*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] = ((2*n + 1 + (6*n + 2)*x + (4*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.