Real-rooted atlas

A347056 → reverse → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
23228666
31201607010360
4400720480140151755
51232280025201120245217938
63584985611200672022403922834020
7998432256443523360015120403258836139968
826880998401612801478408400030240672084045557685
97040029568054912059136040656018480055440105601155552165130
1018022484480017740802196480177408097574436960095040158401540668227494
114526082342912549120076876807138560461260821141126864001544402288020027830705480
121118208633651216400384256256002690688019987968107627524228224120120024024032032254891112842639
1327238401677312047523840820019209609600080720640499699202306304079279202002000360360436803185105409209570
14655360043581440134184960253460480328007680307507200215255040114216960461260801409408032032005241605824039201201466777160
Technical details
Node id
n-d137c81c8a2e6f8d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-511a49fff28ed5ac

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:     7,      3;
  2:    32,     28,      6;
  3:   120,    160,     70,     10;
  4:   400,    720,    480,    140,     15;
  5:  1232,   2800,   2520,   1120,    245,     21;
  6:  3584,   9856,  11200,   6720,   2240,    392,    28;
  7:  9984,  32256,  44352,  33600,  15120,   4032,   588,    36;
  8: 26880,  99840, 161280, 147840,  84000,  30240,  6720,   840,   45;
  9: 70400, 295680, 549120, 591360, 406560, 184800, 55440, 10560, 1155, 55;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+4+(2*n+1)*x)*P(n-1,x) - (2+3*x+x^2)*d/dx 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[((3*n + 4 + (2*n + 1)*x)*P[n - 1, x] - (2 + 3*x + x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((3*n + 4 + (2*n + 1)*x)*v[i - 1] - (2 + 3*x + x^2)*deriv(v[i - 1], x))/(n)); 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.