Real-rooted atlas

A059364 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + 2\,n + 8\,n x + 6\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
2157491180
3105803204317294680
49459852384626663643225159120
5103951376677284941925318254119113399756683040
61351352174286145641015198341210427224911144374249579075334152000
72027025384250953119558131406007523379937045161553628375535767031213190022519380816000
8344594257525508407186078476391879627281334836662942908103203283957208185403074948883441044631110251279133856000
9654729075161937343351779257872441139805608012469152760962612866997732402235132631703002760704789823618897017968395574547110637594655905344000
101374931057538000689461047242593265593478939079013616805022189087855639013656762812786653882528222014057059251256208086729912869112733729893059703504737374888757761784238208000
1131623414322596613623845551341184264497271116681363052581619604428642930624056338007744270666874430798459821319938933757179621828017708497940451687056877956596079337688624574411523481740411754625698560581438720000
1279058535806252646191520693004058240115315690377076792071445962364188466635710631053719683102946088342329875156148126081680306792217057361420559848767226235117562171466926455460146499373652267541867403556206233186020171416705005808762568458936980994560000
132134580466768757769279538740475130477396126998330133870830770284360293622155172808417854712752033020165835317567292125729273497249097742971234307441210288255828256658366211596420910133123023149178295985474831234436213572410694212096032346311634506439900614551354191969545889223757256647319985423360000
1461902833536293752434530405910081504444233248475891945499131195941773225083852944561558861179352162486408544719643962910035394452863027419329170543768527831630568715690234274934705870291337463809340421637841061874016131490165149541631909115786668216277645561336692309694785280570855757605001642223397255011510631741140058401875827257794478338263040000
Technical details
Node id
n-bd5cee0018a9f607 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ff79ef52271940c5

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,           7;
  2:        15,          74,           91;
  3:       105,         803,         2043,          1729;
  4:       945,        9852,        38462,         66636,         43225;
  5:     10395,      137667,       728494,       1925318,       2541191,        1339975;
  6:    135135,     2174286,     14564101,      51983412,     104272249,      111443742,       49579075;
  7:   2027025,    38425095,    311955813,    1406007523,    3799370451,     6155362837,     5535767031,     2131900225;
  8:  34459425,   752550840,   7186078476,   39187962728,  133483666294,   290810320328,   395720818540,   307494888344,   104463111025;
  9: 654729075, 16193734335, 177925787244, 1139805608012, 4691527609626, 12866997732402, 23513263170300, 27607047898236, 18897017968395, 5745471106375;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(8*n+2)*x+(6*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 + (8*n + 2)*x + (6*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 + (8*n + 2)*x + (6*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.