Real-rooted atlas

A131431 → x ↦ x+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(2 + 4\,x + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
228112112252
3804809606402160
4208166449926656332816848
5512512020480409604096016384124416
61216145927296019456029184023347277824886464
72816394242365447884801576960189235212615683604486158592
864001024007168002867200716800011468800114688006553600163840041990400
9143362580482064384963379228901376578027527707033666060288330301447340032282175488
1031744634880571392030474240106659840255983616426639360487587840365690880162529280325058561874451456
1169632153190415319040919142403676569601029439488205887897629412556802941255680196083712078433484814260633612335099904
1215155236372484000972826673152012002918403840933888896217907215363735552192046694401707081728010242490368372454195262075699280540946432
13327680851968010223616074973184037486592001349517312035987128320719742566401079613849601199570944009596567552052344913920174483046402684354560522427760640
14704512197263362564423682051538944112834641924513385676813540157030430948930355254160628121672214170828872214170828852519396966426259698483280799072256115427246083369659056128
Technical details
Node id
n-5245c7f95f442806 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-11aec07ce60b6d61

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:     8,     16;
  2:    28,    112,     112;
  3:    80,    480,     960,     640;
  4:   208,   1664,    4992,    6656,     3328;
  5:   512,   5120,   20480,   40960,    40960,    16384;
  6:  1216,  14592,   72960,  194560,   291840,   233472,    77824;
  7:  2816,  39424,  236544,  788480,  1576960,  1892352,  1261568,   360448;
  8:  6400, 102400,  716800, 2867200,  7168000, 11468800, 11468800,  6553600,  1638400;
  9: 14336, 258048, 2064384, 9633792, 28901376, 57802752, 77070336, 66060288, 33030144, 7340032;
%F Formula
The row polynomials satisfy (3*n-2)*P(n,x) = (6*n+2+(12*n+4)*x)*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[((6*n + 2 + (12*n + 4)*x)*P[n - 1, x])/(3*n - 2)]];
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*n + 2 + (12*n + 4)*x)*v[i - 1])/(3*n - 2)); 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. [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.