Real-rooted atlas

A027466 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + x + n\right)\,P_{n}(x) = \left(32 + 16\,x + 2\,x^{2} + 8\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110212
296404140
389657612081600
48192716823043201618000
5737288192035840768080032200000
6655360884736491520143360230401920642200000
7576716891750406193152229376050176064512448012824000000
8503316489227468873400320330301449175040160563217203210240256260000000
9436207616905969664830472192440401920148635648330301444816896442368230405122800000000
103758096384872415232090596966405536481280220200960059454259211010048013762560110592051200102430000000000
113221225472082678120448959656755206643777536030450647040968884224021799895043460300803784704027033601126402048320000000000
122748779069447730941132809921374453767677254041603986266521601461631057923875536896074742497281038090240100925440648806424576040963400000000000
13233646220902471468255805441005022347264085985245265924990215127040207285859123263337345843214394851328024291311616299892736026240614415335424532480819236000000000000
1419791209299968654209418526721000555581276169380208574464060189671686144279452047114249673340092416253349383372850381979648075572969472839699660866794291235782656114688016384380000000000000
Technical details
Node id
n-956d4c76a0377512 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-76caa1d365acf455

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:        10,         2;
  2:        96,        40,         4;
  3:       896,       576,       120,         8;
  4:      8192,      7168,      2304,       320,        16;
  5:     73728,     81920,     35840,      7680,       800,       32;
  6:    655360,    884736,    491520,    143360,     23040,     1920,      64;
  7:   5767168,   9175040,   6193152,   2293760,    501760,    64512,    4480,    128;
  8:  50331648,  92274688,  73400320,  33030144,   9175040,  1605632,  172032,  10240,   256;
  9: 436207616, 905969664, 830472192, 440401920, 148635648, 33030144, 4816896, 442368, 23040, 512;
%F Formula
The row polynomials satisfy (n+3+x)*P(n,x) = (8*n+32+(2*n+16)*x+2*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[((8*n + 32 + (2*n + 16)*x + 2*x^2)*P[n - 1, x])/(n + 3 + 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] = ((8*n + 32 + (2*n + 16)*x + 2*x^2)*v[i - 1])/(n + 3 + 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.