Real-rooted atlas

basis-map coefficients of A026729 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3916429
42798608193
581544616192161449
62432882540029605603212077
772914896435643744012080153664110309
82187759383347404240882077604412840321281093001
965613840642495056448627231335369891841487361024025611653905
1019683193344218233520453665604344144019126464420940847155225344512132827925
115904997065761315015244447660805685889603355269121013940481643520014246406144010241609465453
12177147486509789399238204265897768714401072054990277762191964544482186496599385604139520146432204820636065809
13531441243609184667676869640272927552871159134568577538252843877523712125810995202103205632206807040116469763440644096278865763897
141594323121910880247224599240375809885360103882090312012893117997768297136207363016337392646512310604885493591046814945283189555279872081923958121912765
15478296960987326563330104122843476738166720121786497050401883869520332815024327910592678720080896018406064934403097827532803277309235221603778568535654418350081638458830135647413
Technical details
Node id
n-906c9612be68275f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,       2;
   3:     9,      16,        4;
   4:    27,      98,       60,        8;
   5:    81,     544,      616,      192,       16;
   6:   243,    2882,     5400,     2960,      560,       32;
   7:   729,   14896,    43564,    37440,    12080,     1536,      64;
   8:  2187,   75938,   334740,   424088,   207760,    44128,    4032,    128;
   9:  6561,  384064,  2495056,  4486272,  3133536,   989184,  148736,  10240,   256;
  10: 19683, 1933442, 18233520, 45366560, 43441440, 19126464, 4209408, 471552, 25344, 512;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + 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, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.