Real-rooted atlas

A013611 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 20\,x + 60\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 300\,x + 240\,x^{2} + 64\,x^{3} + 300\,n x + 400\,n x^{2} + 128\,n x^{3} + 80\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151621
225160144329
31251200216010244509
462580002160020480640057105
531255000018000025600016000036864685989
61562530000013500002560000240000011059202007047932249
778125175000094500002240000028000000193536007024640104857689104941

All 15 rows

Technical details
Node id
n-71be7c5970f096b1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7f34919e5b06c424

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:       5,       16;
  2:      25,      160,       144;
  3:     125,     1200,      2160,       1024;
  4:     625,     8000,     21600,      20480,       6400;
  5:    3125,    50000,    180000,     256000,     160000,      36864;
  6:   15625,   300000,   1350000,    2560000,    2400000,    1105920,     200704;
  7:   78125,  1750000,   9450000,   22400000,   28000000,   19353600,    7024640,   1048576;
  8:  390625, 10000000,  63000000,  179200000,  280000000,  258048000,  140492800,  41943040,   5308416;
  9: 1953125, 56250000, 405000000, 1344000000, 2520000000, 2903040000, 2107392000, 943718400, 238878720, 26214400;
%F Formula
The row polynomials satisfy (25+(60*n-20)*x+16*n^2*x^2)*P(n,x) = (125+(300*n+300)*x+(80*n^2+400*n+240)*x^2+(64*n^2+128*n+64)*x^3)*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[((125 + (300*n + 300)*x + (80*n^2 + 400*n + 240)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(25 + (60*n - 20)*x + 16*n^2*x^2)]];
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] = ((125 + (300*n + 300)*x + (80*n^2 + 400*n + 240)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(25 + (60*n - 20)*x + 16*n^2*x^2)); 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.