Real-rooted atlas

A131431 → D+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(20 - 72\,n + 91\,n^{2} - 48\,n^{3} + 9\,n^{4}\right)\,P_{n}(x) = \left(-2\,n x^{2} - n^{2} x^{2} + 12\,n^{3} x^{2} - 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(20\,n x + 38\,n^{2} x - 60\,n^{3} x + 18\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)
\(P_{2}(x) = 14 + 14\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2141428
30603090
40015652208
500032080400
60000570114684
7000009241541078

All 15 rows

Other recurrences

Derivative · proved equal for all n (proof step)

\(\left(-4 + 2\,x + 12\,n - 5\,n x - 11\,n^{2} + 3\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(-n x + n x^{2} - 2\,n^{2} x + 3\,n^{2} x^{2} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)
Technical details
Node id
n-e1f285b6913430e1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
derived:derivative (equal by e-86702e9cf90939cd)
Fingerprint
f6-2327141074f67bdf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  4;
  2: 14, 14;
  3:  0, 60,  30;
  4:  0,  0, 156,  52;
  5:  0,  0,   0, 320,  80;
  6:  0,  0,   0,   0, 570, 114;
  7:  0,  0,   0,   0,   0, 924,  154;
  8:  0,  0,   0,   0,   0,   0, 1400,  200;
  9:  0,  0,   0,   0,   0,   0,    0, 2016, 252;
%F Formula
The row polynomials satisfy (9*n^4-48*n^3+91*n^2-72*n+20)*P(n,x) = -(9*n^4-12*n^3+n^2+2*n)*x^2*P(n-2,x) + (18*n^4-60*n^3+38*n^2+20*n)*x*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 4, P(2,x) = 14 + 14*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[2, x_] := 14 + 14*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^4 - 12*n^3 + n^2 + 2*n)*x^2*P[n - 2, x] + (18*n^4 - 60*n^3 + 38*n^2 + 20*n)*x*P[n - 1, x])/(9*n^4 - 48*n^3 + 91*n^2 - 72*n + 20)]];
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] = 4; v[3] = 14 + 14*x; for(n = 3, N, my(i = n + 1); v[i] = (-(9*n^4 - 12*n^3 + n^2 + 2*n)*x^2*v[i - 2] + (18*n^4 - 60*n^3 + 38*n^2 + 20*n)*x*v[i - 1])/(9*n^4 - 48*n^3 + 91*n^2 - 72*n + 20)); 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.