Real-rooted atlas

A003506 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(6 - 5\,n + n^{2}\right)\,P_{n}(x) = \left(n + 4\,n x - 9\,n x^{2} - n^{2} - 4\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6\,n - 12\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
\(P_{3}(x) = 6 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
362430
41296108216
5202405403201120
630480162019207504800
742840378067205250151218144
85613447560179202100012096274462720
972201613608403206300054432246964608202752
109028802268080640157500181440123480460807290622080
1111039603564014784034650049896045276025344080190110001830400
12132528053460253440693000119750413582801013760481140132000159725203968
13156686477220411840128700025945923531528329472020849408580002076362246414376960
14182873610810864064022522505189184824023292252167297290400400014534523144963075838764544
Technical details
Node id
n-5ed6ed84468d7a02 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (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,   24;
  4: 12,   96,   108;
  5: 20,  240,   540,   320;
  6: 30,  480,  1620,  1920,   750;
  7: 42,  840,  3780,  6720,  5250,  1512;
  8: 56, 1344,  7560, 17920, 21000, 12096,  2744;
  9: 72, 2016, 13608, 40320, 63000, 54432, 24696, 4608;
%F Formula
The row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^2-n+(4*n^2-4*n)*x-(9*n^2-9*n)*x^2)*P(n-2,x) + ((4*n^2-4*n)*x^2+(4*n^2-4*n)*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+(4*n^2-12*n)*x)*P(n-1,x) - (4*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 6 + 24*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 6 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (4*n^2 - 4*n)*x - (9*n^2 - 9*n)*x^2)*P[n - 2, x] + ((4*n^2 - 4*n)*x^2 + (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + (4*n^2 - 12*n)*x)*P[n - 1, x] - (4*n^2 - 12*n)*x^2*D[P[n - 1, x], x])/(n^2 - 5*n + 6)]];
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 + 24*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n + (4*n^2 - 4*n)*x - (9*n^2 - 9*n)*x^2)*v[i - 2] + ((4*n^2 - 4*n)*x^2 + (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + (4*n^2 - 12*n)*x)*v[i - 1] - (4*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(n^2 - 5*n + 6)); 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.