Real-rooted atlas

A147716 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131518
2990225324
327405202533755832
4811620121504050050625104976
52436075607503037507593757593751889568
67292187027337518225006834375136687501139062534012224
72187765451148175956812547840625143521875239203125170859375612220032
86561262440459270045927000287043750114817500028704375004100625000256289062511019960576
919683885735177147002066715001550036250775018125025833937500553584375006919804687538443359375198359290368
1059049295245066430125885735000775018125046501087500193754531250553584375000103797070312511533007812505766503906253570467226624
11177147974308524357712536536568753653656875025575598125012787799062504567071093750114176777343751902946289062519029462890625864975585937564268410079232
12531441318864608768776501461462750016441455937513153164750007672679437500328829118750001027590996093752283535546875003425303320312503113912109375001297463378906251156831381426176
1315943231036309953108929850569970472507124630906256412167815625427477854375002137389271875008015209769531252226447158203125445289431640625060721286132812505060107177734375194619506835937520822964865671168
14478296933480783010881254475217625089500299234498062529923449806250224425873546875128243356312500056106468386718751870215612890625046755390322265625850098005859375001062622507324218758174019287109375029192926025390625374813367582081024
Technical details
Node id
n-b9fb3905119e37a6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a8e18dd2f0709b33

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:     3,     15;
  2:     9,     90,      225;
  3:    27,    405,     2025,      3375;
  4:    81,   1620,    12150,     40500,      50625;
  5:   243,   6075,    60750,    303750,     759375,     759375;
  6:   729,  21870,   273375,   1822500,    6834375,   13668750,    11390625;
  7:  2187,  76545,  1148175,   9568125,   47840625,  143521875,   239203125,   170859375;
  8:  6561, 262440,  4592700,  45927000,  287043750, 1148175000,  2870437500,  4100625000,  2562890625;
  9: 19683, 885735, 17714700, 206671500, 1550036250, 7750181250, 25833937500, 55358437500, 69198046875, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (3+15*x)*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[(3 + 15*x)*P[n - 1, 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] = (3 + 15*x)*v[i - 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.