Real-rooted atlas

A038271 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(12\,n + 5\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
155
212050170
3216018003754335
4345604320018000250098260
5518400864000540000150000156252088025
674649601555200012960000540000011250009375042595710
7104509440261273600272160000151200000472500007875000546875844814915
81433272320418037760052254720003628800000151200000037800000052500000312500016413546920
9193491763206449725440094058496000783820800004082400000013608000000283500000033750000017578125313909084845
1025798901760096745881600016124313600001567641600000979776000000408240000000113400000000202500000002109375000976562505929393824850
1134054550323201418939596800026605117440000295612416000002155507200000010777536000000374220000000089100000000013921875000012890625000537109375110879664524695
12445805022412802043273019392004256818790400005321023488000004434186240000002586608640000001077753600000003207600000000066825000000009281250000007734375000029296875002056313778457980
1357954652913664028977326456832006640637313024000922310737920000086466631680000005764442112000000280215936000000010007712000000002606175000000004826250000000060328125000004570312500001586914062537870445419934465
1474895243765350404056825703956480010142064259891200015494820397056000016140437913600000012105328435200000067251824640000000280215936000000008756748000000000202702500000000033783750000000038390625000000266601562500085449218750693320462303415590
Technical details
Node id
n-a286a3bdf9075ad0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f055f948fcf55643

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:           5;
  2:         120,          50;
  3:        2160,        1800,         375;
  4:       34560,       43200,       18000,        2500;
  5:      518400,      864000,      540000,      150000,       15625;
  6:     7464960,    15552000,    12960000,     5400000,     1125000,       93750;
  7:   104509440,   261273600,   272160000,   151200000,    47250000,     7875000,     546875;
  8:  1433272320,  4180377600,  5225472000,  3628800000,  1512000000,   378000000,   52500000,   3125000;
  9: 19349176320, 64497254400, 94058496000, 78382080000, 40824000000, 13608000000, 2835000000, 337500000, 17578125;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (12*n+5*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 5*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] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((12*n + 5*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.