Real-rooted atlas

A038208 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 6\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
210015636292
31000309617282166040
4100005553656592164161296139840
5100000948576157824079920014256077763576352
61000000157772164041273631847040964224011664004665699892288
710000000258435456983743488113419353651891840010450944091445762799363019224832
810000000041949672962319296947237659879936244483868167293888000104770713669424128167961698008902400
910000000006771947673653541513216011936344550401057204431360438445840896919578562569896297472513962496100776963395797530112
10100000000001089511627776121854497679363663425553408043106757396480238810602240006860736433152106635460608089187609600372874752060466176124927102412800
11100000000000174921860444162746169646612481098871853561856168543528468480012138829533388804588802372689929661485811507211571008993280773765498880266051174403627970564858266080081920
1210000000000002804749767106566146526338813952324161136876994566389803080065433658667929841664000283917886344069127777254045597696125080767030067211894704588800065049512140801872032808962176782336198955567373012992
13100000000000004493599627370496136906429314170880944530341288468480236702972934844416027301053784704860161658029856232701952574767942177521664119210369151025152151174569586360321169028860313600532571566325761301715836928130606940168551424703362301952
141000000000000007195759403792793630389030426759823362726828813196214272086146192845577912320123406393514910105600926500056575272058883983611213062733824010362809063824588800168277946025885696017253676197376819211061717065957376426283638423552895963609497678364164096384647575242080468992
Technical details
Node id
n-fb4d4dfa4db08258 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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:         10,           6;
  2:        100,         156,           36;
  3:       1000,        3096,         1728,           216;
  4:      10000,       55536,        56592,         16416,          1296;
  5:     100000,      948576,      1578240,        799200,        142560,         7776;
  6:    1000000,    15777216,     40412736,      31847040,       9642240,      1166400,       46656;
  7:   10000000,   258435456,    983743488,    1134193536,     518918400,    104509440,     9144576,     279936;
  8:  100000000,  4194967296,  23192969472,   37659879936,   24448386816,   7293888000,  1047707136,   69424128,   1679616;
  9: 1000000000, 67719476736, 535415132160, 1193634455040, 1057204431360, 438445840896, 91957856256, 9896297472, 513962496, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (10+6*x)*P(n-1,x) + 6*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[(10 + 6*x)*P[n - 1, x] + 6*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] = (10 + 6*x)*v[i - 1] + 6*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.