Real-rooted atlas

A147716 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + n x\right)\,P_{n}(x) = \left(75 + 30\,x + 3\,x^{2} + 15\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115621
222518027432
33375405012151088748
45062581000364506480405174960
575937515187509112502430003037514583464208
611390625273375002050312572900001366875131220510368024448
7170859375478406250430565625191362500478406256889050535815174961326476736
8256289062582012500008611312500459270000014352187502755620003214890020995205904925713241344
93844335937513839609375016607531250010333575000038750906250930021750014467005001417176007971615196830495898225920
10576650390625230660156250031139121093752214337500000968772656250279006525000542512687507085880000597871125295245006495399521245937664
11864975585937538058925781250570883886718754567071093750022835355468750767267943750017902918687502922925500003288291187524357712501071739352125764182093828557824
12129746337890625622782421875000102759099609375091341421875000051379549804687519729747125000053708756062500105225318000001479731034375146146275000964565415038263752069087333470494144278528
13194619506835937510120214355468750182163858398437501781157726562500011132235791015625480912586171875014961724903125003419822835000005770951034062571246309062506269675197503730715820013472029352232052265939388741292032
1429192926025390625163480385742187500318786752197265625340039202343750000233776951611328125112212936773437500392745278707031251025946850500000020198328619218752992344980625003291579478687526115010740001414563081754687309620717445351249377891940270080
Technical details
Node id
n-90989d1b314934e8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6ae5c48b72ea3401

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:          15,            6;
  2:         225,          180,           27;
  3:        3375,         4050,         1215,          108;
  4:       50625,        81000,        36450,         6480,         405;
  5:      759375,      1518750,       911250,       243000,       30375,       1458;
  6:    11390625,     27337500,     20503125,      7290000,     1366875,     131220,       5103;
  7:   170859375,    478406250,    430565625,    191362500,    47840625,    6889050,     535815,     17496;
  8:  2562890625,   8201250000,   8611312500,   4592700000,  1435218750,  275562000,   32148900,   2099520,   59049;
  9: 38443359375, 138396093750, 166075312500, 103335750000, 38750906250, 9300217500, 1446700500, 141717600, 7971615, 196830;
%F Formula
The row polynomials satisfy (5+n*x)*P(n,x) = (75+(15*n+30)*x+(3*n+3)*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[((75 + (15*n + 30)*x + (3*n + 3)*x^2)*P[n - 1, x])/(5 + n*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] = ((75 + (15*n + 30)*x + (3*n + 3)*x^2)*v[i - 1])/(5 + n*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

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.