Real-rooted atlas

A130534 → Veronese 2 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-81 + 18\,x - x^{2} + 108\,n - 12\,n x - 54\,n^{2} + 2\,n^{2} x + 12\,n^{3} - n^{4}\right)\,P_{n-4}(x) + \left(-65 + 10\,x + 76\,n - 4\,n x - 30\,n^{2} + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-25 + 2\,x + 24\,n - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-6 + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
366
435237
522530255
6162435031977
71313239208417136
81181244489816384164664
91172700538650283501801739880
101275357668338604733195280520066040
11150917976919914607912674130680330250953120
121931559552131441367213497069329978521358563383955360
13265967170561991540751223728294596667861245545554648952088640
1439215679782431944321136043227968784147364646013748735300307756315403200
Technical details
Node id
n-01d407861ded857c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-40a9ee76db6f0ae7

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:       1;
  3:       6;
  4:      35,      2;
  5:     225,     30;
  6:    1624,    350,     3;
  7:   13132,   3920,    84;
  8:  118124,  44898,  1638,   4;
  9: 1172700, 538650, 28350, 180;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-12*n^3+54*n^2-108*n+81-(2*n^2-12*n+18)*x+x^2)*P(n-4,x) + (4*n^3-30*n^2+76*n-65-(4*n-10)*x)*P(n-3,x) - (6*n^2-24*n+25-2*x)*P(n-2,x) + (4*n-6)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 6;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 - (2*n^2 - 12*n + 18)*x + x^2)*P[n - 4, x] + (4*n^3 - 30*n^2 + 76*n - 65 - (4*n - 10)*x)*P[n - 3, x] - (6*n^2 - 24*n + 25 - 2*x)*P[n - 2, x] + (4*n - 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] = 1; v[4] = 6; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 - (2*n^2 - 12*n + 18)*x + x^2)*v[i - 4] + (4*n^3 - 30*n^2 + 76*n - 65 - (4*n - 10)*x)*v[i - 3] - (6*n^2 - 24*n + 25 - 2*x)*v[i - 2] + (4*n - 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.