Real-rooted atlas

A178821 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(20 + 12\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x) + \left(12\,x + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1251540
23755851351095
3437513545756094526425
443750242970247590718205670611800
53937503735018621432031468505613303061814081886
6328125051768990132604290104498100316386003827250153090327771570
725781250666278910253621368029240927101337836500269438400235758607217107783939020
819335937581113625454484578081572818908470472732479451410341625020258399701342380603247695189509401125
91396484375945691911457476988662001666891865925147636946957561228276609512841770942013820024790717740595140733454742178191465
1097753906251065040175199119117531422893581141581017042138539212770233446692228756706637860923104240328192087184372275364499635559108049122121122573450
1166650390625116586103274551830332796692407324023828671551123349200974000809057730130650305845470640710643941803354857712122693275515717727525177324147002418056553238053319976475
1244433593750012462511172202027311225431253401440364426553052028392191420212620260682696073800001261549202017104034557134675263205557787988152405285244714750028903835961008318114532096722262088403108549532140
1329052734375001305513563615940397750612509432002744118911458197006876868598448849007931702637646187404817030099894812801669858035134889603463387189690428043920346144474803396348495540001547266025214037818404442037944887402484422704345981280
1418676757812500134393455661657405675685614372867405092832134021249800160893158024577888002304831324192046650017304017169923904420744009418455320160019354367514223170003142886443110414003222427317137028020659701063128407961596274268016733695343401463588514071839696059617079780
Technical details
Node id
n-d4101230aa2f77b9 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         25,          15;
  2:        375,         585,          135;
  3:       4375,       13545,         7560,           945;
  4:      43750,      242970,       247590,         71820,          5670;
  5:     393750,     3735018,      6214320,       3146850,        561330,        30618;
  6:    3281250,    51768990,    132604290,     104498100,      31638600,      3827250,       153090;
  7:   25781250,   666278910,   2536213680,    2924092710,    1337836500,    269438400,     23575860,      721710;
  8:  193359375,  8111362545,  44845780815,   72818908470,   47273247945,  14103416250,   2025839970,   134238060,   3247695;
  9: 1396484375, 94569191145, 747698866200, 1666891865925, 1476369469575, 612282766095, 128417709420, 13820024790, 717740595, 14073345;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+20+(3*n+12)*x)*P(n-1,x) + (3*n+12)*x*d/dx 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[((5*n + 20 + (3*n + 12)*x)*P[n - 1, x] + (3*n + 12)*x*D[P[n - 1, x], x])/(n)]];
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] = ((5*n + 20 + (3*n + 12)*x)*v[i - 1] + (3*n + 12)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.