Real-rooted atlas

A003506 → Veronese 2 (j=1) → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 3\,x - 3\,x^{2} + x^{3}\right)\,P_{n-6}(x) + \left(6 - 12\,x + 6\,x^{2}\right)\,P_{n-5}(x) + \left(-15 + 18\,x - 3\,x^{2}\right)\,P_{n-4}(x) + \left(20 - 12\,x\right)\,P_{n-3}(x) + \left(-15 + 3\,x\right)\,P_{n-2}(x) + \left(6\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
\(P_{3}(x) = 6\)
\(P_{4}(x) = 12 + 8\,x\)
\(P_{5}(x) = 20 + 40\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
222
366
412820
5204060
63012018168
742280126448
856560504321152
972100815122882880
1090168037801440507040
1111026408316528055016896
121323960166321584033007239936
13156572030888411841430093693184
141828008540549609650050655298215040
Technical details
Node id
n-54a374793669f817 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-49176c5aecdbb25b

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:  0;
  1:  0;
  2:  2;
  3:  6;
  4: 12,    8;
  5: 20,   40;
  6: 30,  120,   18;
  7: 42,  280,  126;
  8: 56,  560,  504,  32;
  9: 72, 1008, 1512, 288;
%F Formula
The row polynomials satisfy P(n,x) = -(1-3*x+3*x^2-x^3)*P(n-6,x) + (6-12*x+6*x^2)*P(n-5,x) - (15-18*x+3*x^2)*P(n-4,x) + (20-12*x)*P(n-3,x) - (15-3*x)*P(n-2,x) + 6*P(n-1,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 6, P(4,x) = 12 + 8*x, P(5,x) = 20 + 40*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 6;
P[4, x_] := 12 + 8*x;
P[5, x_] := 20 + 40*x;
P[n_, x_] := P[n, x] = Expand[-(1 - 3*x + 3*x^2 - x^3)*P[n - 6, x] + (6 - 12*x + 6*x^2)*P[n - 5, x] - (15 - 18*x + 3*x^2)*P[n - 4, x] + (20 - 12*x)*P[n - 3, x] - (15 - 3*x)*P[n - 2, x] + 6*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] = 0; v[2] = 0; v[3] = 2; v[4] = 6; v[5] = 12 + 8*x; v[6] = 20 + 40*x; for(n = 6, N, my(i = n + 1); v[i] = -(1 - 3*x + 3*x^2 - x^3)*v[i - 6] + (6 - 12*x + 6*x^2)*v[i - 5] - (15 - 18*x + 3*x^2)*v[i - 4] + (20 - 12*x)*v[i - 3] - (15 - 3*x)*v[i - 2] + 6*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

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.