Real-rooted atlas

A038303 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 - 10\,x + 30\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(1000 + 300\,x + 30\,x^{2} + x^{3} + 300\,n x + 50\,n x^{2} + 2\,n x^{3} + 10\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110414
2100809189
310001200270162486
4100001600054006402532065
51000002000009000016000125036407286
6100000024000001350000320000375002160495109709
710000000280000001890000056000008750007560034306463454094

All 15 rows

Technical details
Node id
n-fded5513452bc5da · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f093baae4d91e0bf

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:         10,          4;
  2:        100,         80,          9;
  3:       1000,       1200,        270,         16;
  4:      10000,      16000,       5400,        640,        25;
  5:     100000,     200000,      90000,      16000,      1250,       36;
  6:    1000000,    2400000,    1350000,     320000,     37500,     2160,      49;
  7:   10000000,   28000000,   18900000,    5600000,    875000,    75600,    3430,     64;
  8:  100000000,  320000000,  252000000,   89600000,  17500000,  2016000,  137200,   5120,   81;
  9: 1000000000, 3600000000, 3240000000, 1344000000, 315000000, 45360000, 4116000, 230400, 7290, 100;
%F Formula
The row polynomials satisfy (100+(30*n-10)*x+n^2*x^2)*P(n,x) = (1000+(300*n+300)*x+(10*n^2+50*n+30)*x^2+(n^2+2*n+1)*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[((1000 + (300*n + 300)*x + (10*n^2 + 50*n + 30)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(100 + (30*n - 10)*x + 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] = ((1000 + (300*n + 300)*x + (10*n^2 + 50*n + 30)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(100 + (30*n - 10)*x + 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.