Real-rooted atlas

A131431 → D → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 18\,x^{2} + 3\,n + 18\,n x + 27\,n x^{2} - 9\,n^{2} - 54\,n^{2} x - 81\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 30\,x - 24\,n - 72\,n x + 18\,n^{2} + 54\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2144256
330180270480
452468140414043328
58096043208640648020480
6114171010260307804617027702116736
715427722079083160187110224532112266630784
8200420037800189000567000102060010206004374003276800
92526048635043810241428840342921651438244408992165337216515072
10310837010044070308031638609491580189831602440692018305190610173081264640
1137411220151470121176063617402290226457255660981525601104216307361442022084326392167424
124441465221978019780201186812049846104149538312320439240480658860480658860288395316786532681862270976
135201872030888030888002084940010007712035026992090069408016888014002251735200202656168011053972802763493208724152320
1460223478422604464864434864830188270082753080328225924098450832922148472153690101665844288318114532415905726695978244640399536128
Technical details
Node id
n-cccb3f6f951d2b32 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f9762973ec1089d7

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,   42;
  3:  30,  180,   270;
  4:  52,  468,  1404,   1404;
  5:  80,  960,  4320,   8640,    6480;
  6: 114, 1710, 10260,  30780,   46170,   27702;
  7: 154, 2772, 20790,  83160,  187110,  224532,  112266;
  8: 200, 4200, 37800, 189000,  567000, 1020600, 1020600,  437400;
  9: 252, 6048, 63504, 381024, 1428840, 3429216, 5143824, 4408992, 1653372;
%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) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
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] = 0; v[2] = 4; 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.