Real-rooted atlas

A038221 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x + n\right)\,P_{n}(x) = \left(12 + 12\,x + 3\,x^{2} + 6\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19312
272549135
3540648243271458
43888648038889728115309
5272165832048600194403645243157464
618662448988852488029160087480131227291594323
71259712391910451438243674160153090036741645927218715943230
8839808030233088470292484115059222044960734832014696641574646561157837977
955427328226748160408146688423263232277766496119042784330674405668704531441196831549681956
10362797056166281984034012224004081466880317447424016665989765952139201417176002125764017714705904915109399071
112358180864119723028482743652736037413446400336721017602095152998491662943682806008480584585100779446805845851177147146444944842
12152374763528489451110421550145126432923832832033672101760024243913267212570917990447140942464126270381602338340400280600848191318765314411412147682405
13979552051205942615777281655442966528280151886643232100737011202626423937280157585436236870037971660822981209451254717165360911952756099485755262178597159432313556617751088
14626913312768411411861504012479493132288231762015313922941594809753626964619089408183849675609609455126174208367699351219210724564410562298120945123482001432034820014322008846984782969129635157244779
Technical details
Node id
n-952f6c3d4cddeec0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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,         3;
  2:       72,        54,         9;
  3:      540,       648,       243,        27;
  4:     3888,      6480,      3888,       972,        81;
  5:    27216,     58320,     48600,     19440,      3645,       243;
  6:   186624,    489888,    524880,    291600,     87480,     13122,      729;
  7:  1259712,   3919104,   5143824,   3674160,   1530900,    367416,    45927,    2187;
  8:  8398080,  30233088,  47029248,  41150592,  22044960,   7348320,  1469664,  157464,   6561;
  9: 55427328, 226748160, 408146688, 423263232, 277766496, 119042784, 33067440, 5668704, 531441, 19683;
%F Formula
The row polynomials satisfy (n+1+x)*P(n,x) = (6*n+12+(3*n+12)*x+3*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[((6*n + 12 + (3*n + 12)*x + 3*x^2)*P[n - 1, x])/(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] = ((6*n + 12 + (3*n + 12)*x + 3*x^2)*v[i - 1])/(n + 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

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.