Real-rooted atlas

A093375 → D → θ+1

palindromicγ-positivenonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} - 2\,n + 7\,n x - 2\,n x^{2} + n^{2} - 4\,n^{2} x + n^{2} x^{2} + n^{3} x\right)\,P_{n}(x) = \left(2\,n x + 2\,n x^{2} + n^{2} + n^{2} x^{3} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2448
3924942
416727216176
52516027016025640
636300720720300362112
749504157522401575504496496
86478430245600560030247846418944
98111525292120961575012096529211528152992
10100162086402352037800378002352086401620100143360
111212200133654224080850997928085042240133652200121377344
121442904198007128015840023284823284815840071280198002904144970752
131693744283141144002895754942085885884942082895751144002831437441692449408
14196473239312176176500500972972134534413453449729725005001761763931247321966078464
Technical details
Node id
n-4869253220535a0b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-98fdf976629e0e14

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: yes
  • γ-positive (gamma_positive): yes
  • 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:  4,    4;
  3:  9,   24,    9;
  4: 16,   72,   72,    16;
  5: 25,  160,  270,   160,    25;
  6: 36,  300,  720,   720,   300,    36;
  7: 49,  504, 1575,  2240,  1575,   504,   49;
  8: 64,  784, 3024,  5600,  5600,  3024,  784,   64;
  9: 81, 1152, 5292, 12096, 15750, 12096, 5292, 1152, 81;
%F Formula
The row polynomials satisfy (n^2-2*n+1+(n^3-4*n^2+7*n-4)*x+(n^2-2*n+1)*x^2)*P(n,x) = (n^2+(n^3+2*n)*x+(n^3+2*n)*x^2+n^2*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (n^3 + 2*n)*x + (n^3 + 2*n)*x^2 + n^2*x^3)*P[n - 1, x])/(n^2 - 2*n + 1 + (n^3 - 4*n^2 + 7*n - 4)*x + (n^2 - 2*n + 1)*x^2)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + (n^3 + 2*n)*x + (n^3 + 2*n)*x^2 + n^2*x^3)*v[i - 1])/(n^2 - 2*n + 1 + (n^3 - 4*n^2 + 7*n - 4)*x + (n^2 - 2*n + 1)*x^2)); 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.