Real-rooted atlas

A094638 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26262860
324164372280840
4120113239926236364015120
572087124206410128812161658240332640
650407466445997615082322775784271838411065608648640
74032070819253224162218532855387376828143206865692824344320259459200
836288073817286560654433272835210531195842130013280268827035219355220806086080008821612800
93628800839779208627538245164266752198475896965078748115286523075360946267906566028069824017041024000335221286400
1039916800103624992012093607584835523028163784157559681173937573248252616574467237231100333763596518190976205615290944052827174400014079294028800
11479001600137921702401803557882881413810291648738176736736026953386581888702278664264961305669557193601697439594264961469554534064647624845984896017961239296000647647525324800
12622702080019702127232028549355466242505269795808014827395656665662351941815686419102375670743044295801482132032703764883416076881909473930643846428581754075648305468912525952066456585395200032382376266240000
138717829120030073786444804784994854553646493519327808030779433102563841466023011686233651684102665314816136550723432820608270359142963532032396179216869332096417638040279634432299908917916659200131491486965708800265826341580800001748648318376960000
141307674368000488593461888008470665098956809031575686629248661613629648032003522550140939595521405651435004802688427067726610084620899270682520642666241756813140047185881623300276929575796608224570695007741685761486845577490197760060528734518966784001143053268797440000101421602465863680000
Technical details
Node id
n-4ae15c45000d2069 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-bd7f1235338ae873

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,        4;
  2:       6,       26,        28;
  3:      24,      164,       372,        280;
  4:     120,     1132,      3992,       6236,        3640;
  5:     720,     8712,     42064,     101288,      121616,       58240;
  6:    5040,    74664,    459976,    1508232,     2775784,     2718384,     1106560;
  7:   40320,   708192,   5322416,   22185328,    55387376,    82814320,    68656928,    24344320;
  8:  362880,  7381728,  65606544,  332728352,  1053119584,  2130013280,  2688270352,  1935522080,   608608000;
  9: 3628800, 83977920, 862753824, 5164266752, 19847589696, 50787481152, 86523075360, 94626790656, 60280698240, 17041024000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(4*n+2)*x+(3*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 + (4*n + 2)*x + (3*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 + (4*n + 2)*x + (3*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.