Real-rooted atlas

A038231 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + x + 4\,n - 2\,n x + n^{2} x\right)\,P_{n}(x) = \left(16\,n + 4\,n x + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28412
348489105
425638414416800
5128025601440320255625
66144153601152038406003637500
7286728601680640358408400100849240625
813107245875251609628672089600161281568641500000
9589824235929630965762064384806400193536282242304819140625
1026214401179648017694720137625606451200193536037632046080324010054687500
111153433657671680973209608650752047308800170311684139520675840712804400121322265625
1250331648276824064519045120519045120324403200136249344397393928110080114048010560058081441875000000
132181038081308622848269903462429989273602108620800101213798434440806484344832148262401830400151008748816910791015625
14939524096610690662413740539904167939932161312030720070849658882755264512787218432166053888256256002818816209664946419661523437500
Technical details
Node id
n-033b1e898256e58f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-af50e1602aec03e4

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      1;
  2:      8,       4;
  3:     48,      48,       9;
  4:    256,     384,     144,      16;
  5:   1280,    2560,    1440,     320,     25;
  6:   6144,   15360,   11520,    3840,    600,     36;
  7:  28672,   86016,   80640,   35840,   8400,   1008,    49;
  8: 131072,  458752,  516096,  286720,  89600,  16128,  1568,   64;
  9: 589824, 2359296, 3096576, 2064384, 806400, 193536, 28224, 2304, 81;
%F Formula
The row polynomials satisfy (4*n-4+(n^2-2*n+1)*x)*P(n,x) = (16*n+(4*n^2+4*n)*x+n^2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + (4*n^2 + 4*n)*x + n^2*x^2)*P[n - 1, x])/(4*n - 4 + (n^2 - 2*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n + (4*n^2 + 4*n)*x + n^2*x^2)*v[i - 1])/(4*n - 4 + (n^2 - 2*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.