Real-rooted atlas

A093375 → D → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
312618
436481296
58018012020400
6150480540240301440
7252105016801260420424704
839220164200448025206725614336
9576352890721260010080453610087241472
108105760176403024031500201607560144090115200
11110089103168064680831606930036960118801980110309760
121452132005346012672019404019958413860063360178202640132811008
13187218876858002316604118405045044324322574001029602574034321562076672
14236626208132132400400810810115315211771768648644504501601603603643681825218304

Other recurrences

Derivative · proved equal for all n (proof step)

\(\left(4 - 2\,x - 4\,n + n x + n^{2}\right)\,P_{n}(x) = \left(-n + n x^{2} + n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
Technical details
Node id
n-1cd5d4ff73ee9007 · revision 0 · JSON
Definition source
derived:one_plus_derivative
derived:derivative (equal by e-d1595534f9f3ca9c)
Fingerprint
f6-e99c3824c4864868

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): yes

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:  12,    6;
  4:  36,   48,   12;
  5:  80,  180,  120,    20;
  6: 150,  480,  540,   240,    30;
  7: 252, 1050, 1680,  1260,   420,   42;
  8: 392, 2016, 4200,  4480,  2520,  672,   56;
  9: 576, 3528, 9072, 12600, 10080, 4536, 1008, 72;
%F Formula
The row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^2-n+(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n-2,x) + (2*n^2-6*n+(2*n^2-6*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 12 + 6*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 12 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*P[n - 2, x] + (2*n^2 - 6*n + (2*n^2 - 6*n)*x)*P[n - 1, 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] = 12 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*v[i - 2] + (2*n^2 - 6*n + (2*n^2 - 6*n)*x)*v[i - 1])/(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.