Real-rooted atlas

A147716 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(225 + 120\,x + 16\,x^{2} + 60\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
119423
234515216513
3607541409126411191
410462597200331204864256240065
5177187520925009720002208002432010245082519
6296156254252500025110000777600013248001167364096106472257
7489796875829237500595350000234360000544320007418880544768163842211156407

All 15 rows

Technical details
Node id
n-88a6b75b97562bd9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-147d292a70e03bef

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:           19,            4;
  2:          345,          152,           16;
  3:         6075,         4140,          912,           64;
  4:       104625,        97200,        33120,         4864,         256;
  5:      1771875,      2092500,       972000,       220800,       24320,        1024;
  6:     29615625,     42525000,     25110000,      7776000,     1324800,      116736,       4096;
  7:    489796875,    829237500,    595350000,    234360000,    54432000,     7418880,     544768,     16384;
  8:   8030390625,  15673500000,  13267800000,   6350400000,  1874880000,   348364800,   39567360,   2490368,    65536;
  9: 130707421875, 289094062500, 282123000000, 159213600000, 57153600000, 13499136000, 2090188800, 203489280, 11206656, 262144;
%F Formula
The row polynomials satisfy (4*n+11+4*x)*P(n,x) = (60*n+225+(16*n+120)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 225 + (16*n + 120)*x + 16*x^2)*P[n - 1, x])/(4*n + 11 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 225 + (16*n + 120)*x + 16*x^2)*v[i - 1])/(4*n + 11 + 4*x)); 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

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.