Real-rooted atlas

A038327 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 3\,n x\right)\,P_{n}(x) = \left(169 + 78\,x + 9\,x^{2} + 39\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113619
216915627352
32197304210531086400
42856152728273785616405114688
53712938568305931901825202632514582031616
64826809133665481156720547455201026675113724510335651584
76274851720272597821052313110796058031142475517444246437317496620756992
88157307213011928816364906760422455800648097043501793806562414739618195845904910737418240
9106044993734404945893460991558524437888112481894708179052468841889417484441064456646908733196830184683593728
10137858491849636269962380991112826015813220780320410520105450136418988888306068244304612645440449067645255879006495393161095929856
111792160394037909866046203415747681568905145363214482208386339297050325131923516487535517869816490207448021405557745182953485092884077212576453876069761024
1223298085122481129035548370664245663832474918251962905102480163533616292475724579715265122275923464614851449447237768348167520559513581220072449580063316191846908733914793674309632
13302875106592253181725063955351837742897898419224258173096231912307079790593647515306746484975665494729507427161449159430530242821680621945940194880654504081326343382802182104811675758772232052215481123719086080
1439373763856992892544150895374925257243395145935817704534094103825445588852188804384530953642891839668125005096293967893767814519378624855909788656923146309365582380185700348623791699990476912106249404807406233500471744535261208778387488768
Technical details
Node id
n-911992e8152204af · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1b244ba6b5adbabe

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:          13,           6;
  2:         169,         156,          27;
  3:        2197,        3042,        1053,         108;
  4:       28561,       52728,       27378,        5616,         405;
  5:      371293,      856830,      593190,      182520,       26325,       1458;
  6:     4826809,    13366548,    11567205,     4745520,     1026675,     113724,      5103;
  7:    62748517,   202725978,   210523131,   107960580,    31142475,    5174442,    464373,     17496;
  8:   815730721,  3011928816,  3649067604,  2245580064,   809704350,  179380656,  24147396,   1819584,   59049;
  9: 10604499373, 44049458934, 60991558524, 43788811248, 18947081790, 5246884188, 941748444, 106445664, 6908733, 196830;
%F Formula
The row polynomials satisfy (13+3*n*x)*P(n,x) = (169+(39*n+78)*x+(9*n+9)*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[((169 + (39*n + 78)*x + (9*n + 9)*x^2)*P[n - 1, x])/(13 + 3*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] = ((169 + (39*n + 78)*x + (9*n + 9)*x^2)*v[i - 1])/(13 + 3*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.