Real-rooted atlas

A038323 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(200 + 360\,x + 162\,x^{2} + 180\,n + 162\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1381856
2112013683242812
32960060480369365832132848
4736000213120021772808864641049766035920
517600000662400009590400065318400199454401889568266897408
640960000019008000003576960000345254400017635968004308215043401222411568334528
7934400000051609600000119750400000150232320000108755136000444426393609047251584612220032493793566976
82099200000001345536000000371589120000057480192000005408363520000313214791680010666233446401861148897281101996057620823636031744
946592000000003400704000000010898841600000020065812480000023279477760000017523097804800084567993753600246847116902403768826516992198359290368869558427699200
1010240000000000083865600000000030606336000000006539304960000000902961561600000083806119936000005256929341440000217460555366400055540601303040075376530339840357046722662436017110075300864
112232320000000000202752000000000008302694400000000020200181760000000032369559552000000035757277839360000027656019578880000014869600137216000053821487453184000122189322866688001492455300728832642684100792321481657996125220864
12483328000000000004821811200000000002189721600000000000597793996800000000010908098150400000000139836497264640000001287262002216960000085338574700544000004014792037048320000129171569887641600026392893739204608029306394996129792115683138142617660597300756782338048
1310403840000000000001130987520000000000056415191040000000000170798284800000000000349709488128000000000510498993438720000000545362339332096000000430313297883955200000249615330999091200000104384592963256320000302261473537081344005614488304521707520571474702424530944208229648656711682465880711110638764032
1422282240000000000000262176768000000000000142504427520000000000047388760473600000000001076029194240000000000017625358201651200000000214409577244262400000001963304421595545600000013554868883344588800000698922926797455360000026304917426740592640006924535575576772608001179042543949558579201107781730853706137637481336758208102499904431751614849925120
Technical details
Node id
n-cd19c3a2e692e922 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5303de72f2074fcb

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:            38,             18;
  2:          1120,           1368,             324;
  3:         29600,          60480,           36936,            5832;
  4:        736000,        2131200,         2177280,          886464,          104976;
  5:      17600000,       66240000,        95904000,        65318400,        19945440,         1889568;
  6:     409600000,     1900800000,      3576960000,      3452544000,      1763596800,       430821504,       34012224;
  7:    9344000000,    51609600000,    119750400000,    150232320000,    108755136000,     44442639360,     9047251584,      612220032;
  8:  209920000000,  1345536000000,   3715891200000,   5748019200000,   5408363520000,   3132147916800,  1066623344640,   186114889728,   11019960576;
  9: 4659200000000, 34007040000000, 108988416000000, 200658124800000, 232794777600000, 175230978048000, 84567993753600, 24684711690240, 3768826516992, 198359290368;
%F Formula
The row polynomials satisfy (9*n+1+9*x)*P(n,x) = (180*n+200+(162*n+360)*x+162*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[((180*n + 200 + (162*n + 360)*x + 162*x^2)*P[n - 1, x])/(9*n + 1 + 9*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] = ((180*n + 200 + (162*n + 360)*x + 162*x^2)*v[i - 1])/(9*n + 1 + 9*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.