Real-rooted atlas

A027466 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112416
214411216272
317282368768644928
4207364480024832460825695232
52488327997446758402099202560010241960960
62985984137912321671577677414401556480135168409642930176
735831808232603648389480448252657664745472001055129668812816384996376576
842998169638649856008720023552762170572830979522566358302726697779234078726553624440930304
951597803526355969638418986041344021780103168011722948608032738377728495452160040422604816515072262144631724310528
10619173642241037594263552405144705433659866664140804153629736960151654603161630931628851235987128320234356736078643200104857617155527540736
11743008370688168491776737288517931814092815988578215526414024829829120065143991959552172015705128962676750286848247065477120131491430403690987524194304488188485304320
12891610044825627255887626240017709830735134724177976044290048456649548077465626456009358704648798325063024641758762935255042157788214067216194207744007178970726417112760321677721614521510131662848
131069932053790724396606421991424365098969753190401073553573570150401445737776388505601029252118709534724225637397037056010554381766230016165293198829158416404372573388810210747875328382990286848785173708867108864450506176577142784
1412839184645488647077367557337907274778436519434649627225681644696371204477487203315875840387190189042591334419329303104171540485912007665306828801149465345497497601448582678839296011871337924526086229044756480020030653726723570191564826843545614546614119005421568
Technical details
Node id
n-34825491574a3572 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         12,           4;
  2:        144,         112,           16;
  3:       1728,        2368,          768,           64;
  4:      20736,       44800,        24832,         4608,          256;
  5:     248832,      799744,       675840,       209920,        25600,        1024;
  6:    2985984,    13791232,     16715776,      7741440,      1556480,      135168,       4096;
  7:   35831808,   232603648,    389480448,    252657664,     74547200,    10551296,     688128,     16384;
  8:  429981696,  3864985600,   8720023552,   7621705728,   3097952256,   635830272,   66977792,   3407872,    65536;
  9: 5159780352, 63559696384, 189860413440, 217801031680, 117229486080, 32738377728, 4954521600, 404226048, 16515072, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (12+4*x)*P(n-1,x) + 4*x*d/dx 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[(12 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = (12 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.