Real-rooted atlas

A038277 → x ↦ x/(x+1) → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(7\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 18\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11818
2252648900
32646136081749633750
4246961905124898884199041125000
52160902222640857304014696640944784035156250
61815156233377201200225603086294403968092802040733441054687500
7148237742287096561470276360504094752097218273609999593856428554022430761718750
81185901922134623456164670952327057326528018147411072027998862796823999025254488159684608878906250000
9933897762192116111041729044999368892231425282858217243840587976118732875596929551365554060130304178523361331224719238281250
1072636492601681015971601729044999360103742699961604001504141376010289582077824017639283561984019439210456064012496635293184035704672266240686645507812500
11559300993021438202553480166420581188401141169699577605135263648099201584595639984896339556208568192049893973503897604811204587875840274925976450048070695251087155218882751464843750
12427102576488120809014492321553258757758401198228184556480616231637771904022184338959788544570454430394562561047773443581849601347137284605235201154689101090201605938401091321036813881976577114112514984130859375000
13323886120503499942002898192141346546956014412115418310515520700963487965540802883964064772510728651892194317532161906947667318966272306473732247691008035025569399736115202701972496551071744126325986851738419227069854325372518413947486877441406250
1424416030622564816193023668568125926923651721921187310994430520967632713535624777603532855979346325632121126490720445450243114681189954311577660068851520547437568858126450293534822408826443488733501030461899733557351825408265284572388650680325247387146149134336375509262084960937500
Technical details
Node id
n-a95087f429b153fe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c53984b4362c13dd

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         0;
  1:        18;
  2:       252,         648;
  3:      2646,       13608,        17496;
  4:     24696,      190512,       489888,       419904;
  5:    216090,     2222640,      8573040,     14696640,       9447840;
  6:   1815156,    23337720,    120022560,    308629440,     396809280,     204073344;
  7:  14823774,   228709656,   1470276360,   5040947520,    9721827360,    9999593856,    4285540224;
  8: 118590192,  2134623456,  16467095232,  70573265280,  181474110720,  279988627968,  239990252544,   88159684608;
  9: 933897762, 19211611104, 172904499936, 889223142528, 2858217243840, 5879761187328, 7559692955136, 5554060130304, 1785233613312;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (7*n+18*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 18.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 18;
P[n_, x_] := P[n, x] = Expand[Together[((7*n + 18*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 18; for(n = 2, N, my(i = n + 1); v[i] = ((7*n + 18*n*x)*v[i - 1])/(n - 1)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.