Real-rooted atlas

A013610 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 3\,n x\right)\,P_{n}(x) = \left(4 + 24\,x + 36\,x^{2} + 12\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
142428
216192432640
36411525184691213312
4256614441472110592103680262144
51024307202764801105920207360014929924980736
640961474561658880884736024883200358318082090188892274688
7163846881289289728619315202322432005016453125852528642866544641677721600
8655363145728495452163963617281857945600535088332893640458249172942848386983526430064771072
9262144141557762548039682378170368133772083204815794995211236854988816511297126413931406950451597803520532575944704
101048576629145601274019840135895449608918138880038526359961611236854988802201506283520278628139008020639121408006810910064649345848836096
11419430427682406462285414407474249728056056872960028252663971849888432390144242165691187204086546038784045406067097600299680042844168916100448256162727720910848
1216777216120795955229896998912398626652160336341237760019373255294976791074591211522324790635397124903855246540807264970735616007192321028259844279728215162881159093058273282814749767106560
1367108864523449139214133126758420728585912321943304929280012592615941734458765541061427220148185506775045100009456402432944446195630080012466689782317056111272933594234886027283903021056149790487530700848413695994232832
142684354562254857830465954591539210552734646272108825076039680783540547485696411358787429990416118548405420032476000882597560321057779739105689601745336569524387842077094760425717761687639492845895688388267301719244819258776968232960828662331436171264
Technical details
Node id
n-4ac80c0f3f62aee0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3151c955f16eaf46

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:      4,       24;
  2:     16,      192,       432;
  3:     64,     1152,      5184,       6912;
  4:    256,     6144,     41472,     110592,      103680;
  5:   1024,    30720,    276480,    1105920,     2073600,     1492992;
  6:   4096,   147456,   1658880,    8847360,    24883200,    35831808,     20901888;
  7:  16384,   688128,   9289728,   61931520,   232243200,   501645312,    585252864,    286654464;
  8:  65536,  3145728,  49545216,  396361728,  1857945600,  5350883328,   9364045824,   9172942848,   3869835264;
  9: 262144, 14155776, 254803968, 2378170368, 13377208320, 48157949952, 112368549888, 165112971264, 139314069504, 51597803520;
%F Formula
The row polynomials satisfy (1+3*n*x)*P(n,x) = (4+(12*n+24)*x+(36*n+36)*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[((4 + (12*n + 24)*x + (36*n + 36)*x^2)*P[n - 1, x])/(1 + 3*n*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] = ((4 + (12*n + 24)*x + (36*n + 36)*x^2)*v[i - 1])/(1 + 3*n*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.