Real-rooted atlas

A038243 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2483280
34325761921200
4345669124608102416000
5259206912069120307205120200000
6186624622080829440552960184320245762400000
713063685225472870912077414403870720103219211468828000000
8895795241803776836075529289728061931520247726085505024524288320000000
96046617632248627275246796810032906248360755204459069441486356482831155223592963600000000
10403107840241864704064497254401003290624010032906240668860416029727129608493465601415577601048576040000000000
112660511744177367449605321023488094595973120110361968640882895749124904976384018685624320467140608069206016046137344440000000000
1217414258688127704563712425681879040851363758080113515167744010594748989447063165992963363412377601121137459202491416576033218887682013265924800000000000
131131926814729055414517763320318656512737848590336011067728855040118055774453769182115790848524692330905621862180454406477683097601295536619521570347417687241523252000000000000
14731398864896633879016243225355160649728619792815882241032988026470401239585631764481101853894901767345692632678436728463163392136031345049603627502534656659545915392732828794883758096384560000000000000
Technical details
Node id
n-450d83dfd8c67163 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f06e51c139cfae05

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:       48,        32;
  3:      432,       576,       192;
  4:     3456,      6912,      4608,       1024;
  5:    25920,     69120,     69120,      30720,      5120;
  6:   186624,    622080,    829440,     552960,    184320,     24576;
  7:  1306368,   5225472,   8709120,    7741440,   3870720,   1032192,    114688;
  8:  8957952,  41803776,  83607552,   92897280,  61931520,  24772608,   5505024,   524288;
  9: 60466176, 322486272, 752467968, 1003290624, 836075520, 445906944, 148635648, 28311552, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (6*n+4*n*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[((6*n + 4*n*x)*P[n - 1, x])/(n - 1)]];
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] = ((6*n + 4*n*x)*v[i - 1])/(n - 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.