Real-rooted atlas

A038208 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
23628468
3216296968616
4129628001552288165952
57776249922112065608003261280
64665621548826118412096024320211264670784
72799361817216304281619738885824008243253761287784192
816796161509760034062592297722881210137624837122616321331225695472384
910077696124140032370821120425392640228963840639421449676800789504322565121233836544
10604661761013275648395649126458463539204056279040148100198430206668835143680228864076800102416753444864
113627970568227137536415914639367806922956868480614400318085898248399204352130700697612063744064204801802242048238373283840
12217678233666542694400432368914432102001368268811148670607366458986659842148028579844293854822452680376323953664001752678441779240963545290559488
131306069401653669511987244567745331201310490202112017648166215680125641127772165158248775680128837668044820177392435220024868864124642918446751744958464819254993429757952
1478364164096431968234700845641135570944166172373319680273284131061760236322136866816117976703516672360840311603207015779696640884144701440724568965123801907200122257408217907216384887854865661952
Technical details
Node id
n-ffc3f9478b0d8c7f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:        6,         2;
  2:       36,        28,         4;
  3:      216,       296,        96,         8;
  4:     1296,      2800,      1552,       288,        16;
  5:     7776,     24992,     21120,      6560,       800,       32;
  6:    46656,    215488,    261184,    120960,     24320,     2112,      64;
  7:   279936,   1817216,   3042816,   1973888,    582400,    82432,    5376,    128;
  8:  1679616,  15097600,  34062592,  29772288,  12101376,  2483712,  261632,  13312,   256;
  9: 10077696, 124140032, 370821120, 425392640, 228963840, 63942144, 9676800, 789504, 32256, 512;
%F Formula
The row polynomials satisfy P(n,x) = (6+2*x)*P(n-1,x) + 2*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[(6 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = (6 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 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

  • 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.