Real-rooted atlas

A038279 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + x + n\right)\,P_{n}(x) = \left(64 + 16\,x + x^{2} + 8\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19110
28018199
3704240271972
4614428164803619477
55324830720704080045191854
645875231948892160140801200541885735
73932160321126411182082150402464016806318503056
8335544323145728012845056298188843008039424224072181310473
9285212672301989888141557760385351686709248774144591362880811774840978
10241591910428521267201509949440471859200963379201341849612902408448036009017360989291
1120401094656265751101441568669696055364812801297612800211943424246005762027520116160440099169735688020
121717986918402448131358721594506608646274678784016609443840311427072042388684842172416304128015488052801081659002251789
13144310901145622333829939201591285383168690952863744203927060480431845539846747586560787218432685301764392960201344624011716213449802582
141209462790553620203526160384156336809574407425998454784241833502310457099576934410076395929613495173120137763225610660249661501442562567280126158462914051567
Technical details
Node id
n-df60d2271b42ad59 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b2d2fcc245b18987

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:         9,         1;
  2:        80,        18,         1;
  3:       704,       240,        27,        1;
  4:      6144,      2816,       480,       36,       1;
  5:     53248,     30720,      7040,      800,      45,      1;
  6:    458752,    319488,     92160,    14080,    1200,     54,     1;
  7:   3932160,   3211264,   1118208,   215040,   24640,   1680,    63,    1;
  8:  33554432,  31457280,  12845056,  2981888,  430080,  39424,  2240,   72,  1;
  9: 285212672, 301989888, 141557760, 38535168, 6709248, 774144, 59136, 2880, 81, 1;
%F Formula
The row polynomials satisfy (n+7+x)*P(n,x) = (8*n+64+(n+16)*x+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[((8*n + 64 + (n + 16)*x + x^2)*P[n - 1, x])/(n + 7 + 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] = ((8*n + 64 + (n + 16)*x + x^2)*v[i - 1])/(n + 7 + 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.