Real-rooted atlas

A106246 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{6}\right)\,P_{n-6}(x) + \left(6\,x^{5}\right)\,P_{n-5}(x) + \left(-15\,x^{4}\right)\,P_{n-4}(x) + \left(20\,x^{3}\right)\,P_{n-3}(x) + \left(-15\,x^{2}\right)\,P_{n-2}(x) + \left(6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)
\(P_{2}(x) = 1 + 32\,x + 27\,x^{2}\)
\(P_{3}(x) = 24\,x + 162\,x^{2} + 64\,x^{3}\)
\(P_{4}(x) = 162\,x^{2} + 512\,x^{3} + 125\,x^{4}\)
\(P_{5}(x) = 640\,x^{3} + 1250\,x^{4} + 216\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
21322760
302416264250
400162512125799
500064012502162106
60000187525923434810
700000453648025129850

All 15 rows

Technical details
Node id
n-3c7bd17b6d3d5ce9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-abf88017af2e1bb7

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: 2,  8;
  2: 1, 32,  27;
  3: 0, 24, 162,  64;
  4: 0,  0, 162, 512,  125;
  5: 0,  0,   0, 640, 1250,  216;
  6: 0,  0,   0,   0, 1875, 2592,  343;
  7: 0,  0,   0,   0,    0, 4536, 4802,   512;
  8: 0,  0,   0,   0,    0,    0, 9604,  8192,   729;
  9: 0,  0,   0,   0,    0,    0,    0, 18432, 13122, 1000;
%F Formula
The row polynomials satisfy P(n,x) = -x^6*P(n-6,x) + 6*x^5*P(n-5,x) - 15*x^4*P(n-4,x) + 20*x^3*P(n-3,x) - 15*x^2*P(n-2,x) + 6*x*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2 + 8*x, P(2,x) = 1 + 32*x + 27*x^2, P(3,x) = 24*x + 162*x^2 + 64*x^3, P(4,x) = 162*x^2 + 512*x^3 + 125*x^4, P(5,x) = 640*x^3 + 1250*x^4 + 216*x^5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[2, x_] := 1 + 32*x + 27*x^2;
P[3, x_] := 24*x + 162*x^2 + 64*x^3;
P[4, x_] := 162*x^2 + 512*x^3 + 125*x^4;
P[5, x_] := 640*x^3 + 1250*x^4 + 216*x^5;
P[n_, x_] := P[n, x] = Expand[-x^6*P[n - 6, x] + 6*x^5*P[n - 5, x] - 15*x^4*P[n - 4, x] + 20*x^3*P[n - 3, x] - 15*x^2*P[n - 2, x] + 6*x*P[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] = 1; v[2] = 2 + 8*x; v[3] = 1 + 32*x + 27*x^2; v[4] = 24*x + 162*x^2 + 64*x^3; v[5] = 162*x^2 + 512*x^3 + 125*x^4; v[6] = 640*x^3 + 1250*x^4 + 216*x^5; for(n = 6, N, my(i = n + 1); v[i] = -x^6*v[i - 6] + 6*x^5*v[i - 5] - 15*x^4*v[i - 4] + 20*x^3*v[i - 3] - 15*x^2*v[i - 2] + 6*x*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.