Real-rooted atlas

A038303 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2661884
31089594811764
41597213068356432432928
521961523958098010178201215576240
62898918395307021562205880608019043749680832
737202781608772784150723515093540308731533679815309158120256
846769210489286674473052733633205788090561240148191121347192524882529924096
9578768978712627686808120537010446574746024224139069048903069666686004519631217714739846304512
10707384307301736306936101894153021201205370104404931059518013448344140244515348028579716019486170590490619831403520
11855935011833233436821409028649064445652083568323320994430336130325449928188739658927701152715212011789132857144929019486179545403614208
121027122014199630813660425988420186278536203437887733478018752114909880715989842013619526995691283803960199605187218454047156531402572174446377292145784346107904
131223987066921194005775855378446008663783067665462421620970603351940540141051462664962970644653933973088410879326170856185443059730522477946634018391047246911952756207261992211062582636544
1414499539100451025140745681068998841212929629472484121292962947245735542702019130281563005371848210238654740794482792360383853045711646239699486540094540909440737586287022036221231918346466696156633336020476674048
Technical details
Node id
n-2de16a679fd7052d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4f0d71d8efeb5cf0

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:          3;
  2:         66,          18;
  3:       1089,         594,          81;
  4:      15972,       13068,        3564,        324;
  5:     219615,      239580,       98010,      17820,       1215;
  6:    2898918,     3953070,     2156220,     588060,      80190,      4374;
  7:   37202781,    60877278,    41507235,   15093540,    3087315,    336798,    15309;
  8:  467692104,   892866744,   730527336,  332057880,   90561240,  14819112,  1347192,   52488;
  9: 5787689787, 12627686808, 12053701044, 6574746024, 2241390690, 489030696, 66686004, 5196312, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 3*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 3*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.