Real-rooted atlas

A013611 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 80\,x + 240\,n x + 256\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 1200\,x + 3840\,x^{2} + 4096\,x^{3} + 1200\,n x + 6400\,n x^{2} + 8192\,n x^{3} + 1280\,n^{2} x^{2} + 4096\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
156469
22564023042969
312548003456065536105021
462532000345600131072016384003327345
53125200000288000016384000409600003774873698175861
61562512000002160000016384000061440000011324620808220835842755601289
77812570000001512000001433600000716800000019818086400287729254401717986918474530759149

All 15 rows

Technical details
Node id
n-a06d487b0a5cfe40 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8d44fb57f80b5d40

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,        64;
  2:      25,       640,       2304;
  3:     125,      4800,      34560,       65536;
  4:     625,     32000,     345600,     1310720,      1638400;
  5:    3125,    200000,    2880000,    16384000,     40960000,      37748736;
  6:   15625,   1200000,   21600000,   163840000,    614400000,    1132462080,     822083584;
  7:   78125,   7000000,  151200000,  1433600000,   7168000000,   19818086400,   28772925440,    17179869184;
  8:  390625,  40000000, 1008000000, 11468800000,  71680000000,  264241152000,  575458508800,   687194767360,   347892350976;
  9: 1953125, 225000000, 6480000000, 86016000000, 645120000000, 2972712960000, 8631877632000, 15461882265600, 15655155793920, 6871947673600;
%F Formula
The row polynomials satisfy (25+(240*n-80)*x+256*n^2*x^2)*P(n,x) = (125+(1200*n+1200)*x+(1280*n^2+6400*n+3840)*x^2+(4096*n^2+8192*n+4096)*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 + (1200*n + 1200)*x + (1280*n^2 + 6400*n + 3840)*x^2 + (4096*n^2 + 8192*n + 4096)*x^3)*P[n - 1, x])/(25 + (240*n - 80)*x + 256*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 + (1200*n + 1200)*x + (1280*n^2 + 6400*n + 3840)*x^2 + (4096*n^2 + 8192*n + 4096)*x^3)*v[i - 1])/(25 + (240*n - 80)*x + 256*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

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.