Real-rooted atlas

A008277 → θ+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(6\,x - 18\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(3\,x - 9\,x^{2}\right)\,P_{n-3}'(x) + \left(x^{2} - 4\,x^{3}\right)\,P_{n-3}''(x) + \left(-6 + 12\,x - 12\,x^{2}\right)\,P_{n-2}(x) + \left(-2 + 4\,x\right)\,P_{n-2}'(x) + \left(-x + 4\,x^{2}\right)\,P_{n-2}''(x) + \left(6 + 12\,x\right)\,P_{n-1}(x) + \left(1\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2145
3112922
4128541699
516022516025471
611248101040375362386
712522709560035007564912867

All 15 rows

Technical details
Node id
n-dce4c0707068cc4f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-915ca4b22aed877c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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: 1;
  2: 1,    4;
  3: 1,   12,     9;
  4: 1,   28,    54,     16;
  5: 1,   60,   225,    160,     25;
  6: 1,  124,   810,   1040,    375,    36;
  7: 1,  252,  2709,   5600,   3500,   756,    49;
  8: 1,  508,  8694,  27216,  26250,  9576,  1372,   64;
  9: 1, 1020, 27225, 124320, 173775, 95256, 22638, 2304, 81;
%F Formula
The row polynomials satisfy 4*P(n,x) = (6*x-18*x^2+4*x^3)*P(n-3,x) + (3*x-9*x^2)*d/dx P(n-3,x) + (x^2-4*x^3)*d^2/dx^2 P(n-3,x) - (6-12*x+12*x^2)*P(n-2,x) - (2-4*x)*d/dx P(n-2,x) - (x-4*x^2)*d^2/dx^2 P(n-2,x) + (6+12*x)*P(n-1,x) + d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*x - 18*x^2 + 4*x^3)*P[n - 3, x] + (3*x - 9*x^2)*D[P[n - 3, x], x] + (x^2 - 4*x^3)*D[P[n - 3, x], {x, 2}] - (6 - 12*x + 12*x^2)*P[n - 2, x] - (2 - 4*x)*D[P[n - 2, x], x] - (x - 4*x^2)*D[P[n - 2, x], {x, 2}] + (6 + 12*x)*P[n - 1, x] + D[P[n - 1, x], x])/(4)]];
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] = 1; v[3] = 1 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = ((6*x - 18*x^2 + 4*x^3)*v[i - 3] + (3*x - 9*x^2)*deriv(v[i - 3], x) + (x^2 - 4*x^3)*deriv(deriv(v[i - 3], x), x) - (6 - 12*x + 12*x^2)*v[i - 2] - (2 - 4*x)*deriv(v[i - 2], x) - (x - 4*x^2)*deriv(deriv(v[i - 2], x), x) + (6 + 12*x)*v[i - 1] + deriv(v[i - 1], x))/(4)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.