Real-rooted atlas

A013611 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 - 4\,x - 11\,n + 4\,n x + 4\,n^{2}\right)\,P_{n}(x) = \left(-3\,n - 8\,n x + 16\,n x^{2} + 4\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2403272
3108480192780
42081728384010246800
534041601728025600512052500
650481604992013824015360024576375000
7700141121142404659209676808601601146882537500
892822400225792121856037273606193152458752052428816500000
91188334084032002709504109670402683699237158912235929602359296104062500
101480475206681605376000270950408773632017891328021233664011796480010485760640625000
11180465120104544097996805913600023843635264339968011245977601167851520576716800461373443867187500
122160865921562880167270401175961605677056001907490816441188352067475865606228541440276824064020132659222968750000
132548112320225139227089920217451520122300006449201152001416993177628677242880389860556803238841548813086228480872415232134570312500
1429681426883144960420259843792588802435457024114146672643936092160099189522432178436177920218321911808164886478848610690662403758096384779296875000
Technical details
Node id
n-369cec4fbc46cd80 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-236e6d52004bfa9f

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:    4;
  2:   40,    32;
  3:  108,   480,    192;
  4:  208,  1728,   3840,    1024;
  5:  340,  4160,  17280,   25600,     5120;
  6:  504,  8160,  49920,  138240,   153600,    24576;
  7:  700, 14112, 114240,  465920,   967680,   860160,   114688;
  8:  928, 22400, 225792, 1218560,  3727360,  6193152,  4587520,   524288;
  9: 1188, 33408, 403200, 2709504, 10967040, 26836992, 37158912, 23592960, 2359296;
%F Formula
The row polynomials satisfy (4*n^2-11*n+7+(4*n-4)*x)*P(n,x) = (4*n^2-3*n+(16*n^2-8*n)*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 3*n + (16*n^2 - 8*n)*x + 16*n*x^2)*P[n - 1, x])/(4*n^2 - 11*n + 7 + (4*n - 4)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 3*n + (16*n^2 - 8*n)*x + 16*n*x^2)*v[i - 1])/(4*n^2 - 11*n + 7 + (4*n - 4)*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.