Real-rooted atlas

A027467 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(15\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
230232
3675903768
4135002700180416384
52531256750067503005327680
6455625015187502025001350045066291456
779734375318937505315625472500236256307117440512
81366875000637875000127575000141750009450003780084082147483648
923066015625123018750002870437500382725000318937501701000567001080938654705664
1038443359375023066015625061509375000956812500095681250063787500283500081000135010687194767360
11634315429687542287695312501268630859375225534375000263123437502104987500116943750445500011137516501112094627905536
12103797070312500761178515625002537261718750050745234375006766031250006314962500042099750002004750006682500148500198012211106232532992
1316867023925781251349361914062500494766035156250109948007812500164922011718751759168125000136824187500781852500032577187596525001930502340133659174697238528
14272467309570312502361383349609375094455333984375002308908164062500384818027343750461781632812504104725625000273648375000136824187505067562501351350024570027301463050394783186944
Technical details
Node id
n-236c1003708dee15 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8deca67d09eebe9e

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:           1;
  2:          30,           2;
  3:         675,          90,          3;
  4:       13500,        2700,        180,         4;
  5:      253125,       67500,       6750,       300,        5;
  6:     4556250,     1518750,     202500,     13500,      450,       6;
  7:    79734375,    31893750,    5315625,    472500,    23625,     630,     7;
  8:  1366875000,   637875000,  127575000,  14175000,   945000,   37800,   840,    8;
  9: 23066015625, 12301875000, 2870437500, 382725000, 31893750, 1701000, 56700, 1080, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (15*n+n*x)*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[((15*n + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((15*n + 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.