Real-rooted atlas

A038335 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(147 + 126\,x + 27\,x^{2} + 63\,n + 27\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
130939
2819540811440
32116822113729072951300
45278777620483980348748065611782000
512835746237544651714608059705109841505904960750000
630630757569313028464137055530862944080601885106288205314412041200000
7720435416419297377225218336039461346878165548609136801015583751111602610478296967797000000
816750123431351871349980869470558010052400649470424243806979069997156992121870050121147912560430467212230740000000
938579316548221356759997935321007896742224187570506627001061113151775639274967305989449616193921410210579961162261467038742048972827100000000
10881650851704913472138489339806105419990708856302369022667204220336399107501910003673196085891245095897012149506535040158648690245511622614670034867844012361960000000000
1120015857173841208728343431878609171870855222320102014788596933920515598363331101320835626607023285031515060607735328331903778482901503501433711201745135592700511506388523303138105960976173210000000000
124518579756994650921617125747748496047133054532144488661873507880035236054399292117215853533692464795178851215041278926419130048623236366220208112480701009519151804201720453440188474644011654112971814592402824295364812444628600000000000
13101510419657135645852867383156837415531264601856243287016018381891267536350554180980010549103065301272943435542850971965700306350598759842514042334844340300711114831855970542146224911312374895211091601293052482004684849942138110147519227590254186582832978121827000000000000
14227074395276505607671279031287679909137083330645138880757178395311327796221805467205790295749273950424514560696265837397245562673181214639987040991182605514310777677123959616677379835972599556076459835875881842433882535923677241795834208405568210491909243924491067583647898180228767924549612487143880000000000000
Technical details
Node id
n-6f343d3feaf6bcf9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3150a1751f9b8c12

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:            30,              9;
  2:           819,            540,             81;
  3:         21168,          22113,           7290,            729;
  4:        527877,         762048,         398034,          87480,           6561;
  5:      12835746,       23754465,       17146080,        5970510,         984150,         59049;
  6:     306307575,      693130284,      641370555,      308629440,       80601885,      10628820,       531441;
  7:    7204354164,    19297377225,    21833603946,    13468781655,     4860913680,    1015583751,    111602610,      4782969;
  8:  167501234313,   518713499808,   694705580100,   524006494704,   242438069790,   69997156992,  12187005012,   1147912560,    43046721;
  9: 3857931654822, 13567599979353, 21007896742224, 18757050662700, 10611131517756, 3927496730598, 944961619392, 141021057996, 11622614670, 387420489;
%F Formula
The row polynomials satisfy (3*n+4+3*x)*P(n,x) = (63*n+147+(27*n+126)*x+27*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[((63*n + 147 + (27*n + 126)*x + 27*x^2)*P[n - 1, x])/(3*n + 4 + 3*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] = ((63*n + 147 + (27*n + 126)*x + 27*x^2)*v[i - 1])/(3*n + 4 + 3*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

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.