Real-rooted atlas

A131431 → θ+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 18\,x^{2} + 3\,n + 18\,n x + 27\,n x^{2} - 9\,n^{2} - 54\,n^{2} x - 81\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 30\,x - 24\,n - 72\,n x + 18\,n^{2} + 54\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
182432
221126189336
340360108010802560
46578035107020526516640
5961440864025920388802332898304
61332394179557182016159519391496957544768
71763696332641663204989608981288981283849122883584

All 15 rows

Technical details
Node id
n-321a3c7f3ffdd8bf · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f9762973ec1089d7

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:   1;
  1:   8,   24;
  2:  21,  126,   189;
  3:  40,  360,  1080,   1080;
  4:  65,  780,  3510,   7020,    5265;
  5:  96, 1440,  8640,  25920,   38880,   23328;
  6: 133, 2394, 17955,  71820,  161595,  193914,    96957;
  7: 176, 3696, 33264, 166320,  498960,  898128,   898128,   384912;
  8: 225, 5400, 56700, 340200, 1275750, 3061800,  4592700,  3936600,  1476225;
  9: 280, 7560, 90720, 635040, 2857680, 8573040, 17146080, 22044960, 16533720, 5511240;
%F Formula
The row polynomials satisfy (9*n^2-21*n+10)*P(n,x) = -(9*n^2-3*n-2+(54*n^2-18*n-12)*x+(81*n^2-27*n-18)*x^2)*P(n-2,x) + (18*n^2-24*n-10+(54*n^2-72*n-30)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 3*n - 2 + (54*n^2 - 18*n - 12)*x + (81*n^2 - 27*n - 18)*x^2)*P[n - 2, x] + (18*n^2 - 24*n - 10 + (54*n^2 - 72*n - 30)*x)*P[n - 1, x])/(9*n^2 - 21*n + 10)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 3*n - 2 + (54*n^2 - 18*n - 12)*x + (81*n^2 - 27*n - 18)*x^2)*v[i - 2] + (18*n^2 - 24*n - 10 + (54*n^2 - 72*n - 30)*x)*v[i - 1])/(9*n^2 - 21*n + 10)); 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.