Real-rooted atlas

A013611 → basis A007318 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191625
281288256625
372938886912409615625
465614665612441614745665536390625
55904952488018662403317760294912010485769765625
653144156687042519424059719680796262405662310416777216244140625
74782969595213923174474249405849601672151040178362777610569646082684354566103515625
843046721612220032380936908813544423424300987187204280706662438050725888193273528324294967296152587890625
93874204896198727824440798423041828497162244875992432648668430991361027369598976782757789696347892350976687194767363814697265625
1034867844016198727824049589822592023509249228807313988648960156031757844482311581597696023482733690880156551557939206184752906240109951162777695367431640625
1131381059609613674054576545488048512029092695920640103440696606720257452400443392457693156343808581197658849280516620141199360306145268858880108851651149824175921860444162384185791015625
122824295364816025163444928589127092392963491123510476801396449404190720397212274969804882384768141885441255386943114444813948743812382720110212296789196805877989162090496189995609279692828147497671065659604644775390625
13254186582832958745343588048626616998272512408461450725785618153842254479360580922952143339521377002553228656642448004539073167363264006052097556483223709681084006402292415773215293441111474314286202883293257227514675245035996273704961490116119384765625
142287679245496156937794554569665794784818613764678740253768089622873841240643993681329213300067532821687790213351342084406408170331701248685441270940486860881237483963316961287221109685628174336466819212000205209620747520533342453765674535530486824967205759403792793637252902984619140625
Technical details
Node id
n-6e6aac684bd07a75 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-f3c17ca3ebbc301e

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:         9,         16;
  2:        81,        288,         256;
  3:       729,       3888,        6912,         4096;
  4:      6561,      46656,      124416,       147456,        65536;
  5:     59049,     524880,     1866240,      3317760,      2949120,      1048576;
  6:    531441,    5668704,    25194240,     59719680,     79626240,     56623104,      16777216;
  7:   4782969,   59521392,   317447424,    940584960,   1672151040,   1783627776,    1056964608,    268435456;
  8:  43046721,  612220032,  3809369088,  13544423424,  30098718720,  42807066624,   38050725888,  19327352832,   4294967296;
  9: 387420489, 6198727824, 44079842304, 182849716224, 487599243264, 866843099136, 1027369598976, 782757789696, 347892350976, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (9+16*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[(9 + 16*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] = (9 + 16*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.