Real-rooted atlas

A136124 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 7\,x + 5\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
212454299
3603215703361287
4360246663097146302419305
5252020862688231131129262830240328185
6201601946167800661661949198525612608283326406235515
718144019934649355986243183333781069235170524181236963991680130945815
8181440022293360119474892364811148694245249843244236638453772275524848518918403011753745
9199584002706285601626330852568557111612744053811189951200821882841079611969126136442815811272648576075293843625
102395008003546918720235753986249262181617223821221247241910224814951086773078242605567557223267478938475140200800108972864002032933777875
11311351040049941956160363230881632158128998822045787078208889259724625489133469164705501371260747494898416630711444699619240544134390793600017435658240058955079558375
124358914560075211706304059342455975682831298482282490983839292172207474177711942344272149221013418874084648622370897610070132233102941577064987082381932482528742706560029640619008001827607466309625
1365383718400012066360566400102551791098240531511193098584187439131619341247498217729382268898617437130151124800099557075641310319767472717610172701104918336567647652128587221560596934624644996346156928005335311421440060311046388217625
141046139494400020548467555840018700895083334401045266312044390440088973727967688111610583374686444232624492679908710368753890596793893446771351954078532411723973498509720284104945334022336142350008999830992489592880268716161034670752559360010137091700736002110886623587616875
Technical details
Node id
n-e462c7a5b4c0379b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-778f5ea1bb6e399b

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,         6;
  2:       12,        45,         42;
  3:       60,       321,        570,        336;
  4:      360,      2466,       6309,       7146,        3024;
  5:     2520,     20862,      68823,     113112,       92628,       30240;
  6:    20160,    194616,     780066,    1661949,     1985256,     1260828,      332640;
  7:   181440,   1993464,    9355986,   24318333,    37810692,    35170524,    18123696,     3991680;
  8:  1814400,  22293360,  119474892,  364811148,   694245249,   843244236,   638453772,   275524848,   51891840;
  9: 19958400, 270628560, 1626330852, 5685571116, 12744053811, 18995120082, 18828410796, 11969126136, 4428158112, 726485760;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(2*n+7)*x+(n+5)*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 + (2*n + 7)*x + (n + 5)*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 + 2 + (2*n + 7)*x + (n + 5)*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

Same fingerprint ?

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.