Real-rooted atlas

A126351 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 2\,x + 2\,n x + n x^{2} - 2\,n^{2} - 2\,n^{2} x - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 2\,x + x^{2} + 2\,n + 5\,n x + 4\,n x^{2} + n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 2\,n + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
213821
3718624181
4383692387641526
5212951324347148416013252
6123603738242732225445245384120647
775242274486397002286444105555177308961157355
8480513205678436107793326684170162546582858471204811702732
9321611315825164327552513687415624322615939309619841151902324608124565350
1022524511125413070299548752399595508324328615163218138494932068272448614385102401393018873
111647768751025159368277925079042952481044152850235259061926810351266242530858963406193119762702252816332177889
1212567640528647737100262633915804621676488451979845770388378410481938512275063193438961269574011139303460634146749152200322079170
13997576396675274190292253379497368500794380948643419584660561585120960337636231926138605046016375931181256293256980568004283203115241064962565144613544
14822673935776758526798162499258541530548674866683279385146411957941558108204560168091182027862148897529596891883062197677167503098146092823147940006494749322937634224914169579
Technical details
Node id
n-aa7e5e0bfc2517f9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4dd5df82d10aa6b7

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       2;
  2:      13,        8;
  3:      71,       86,       24;
  4:     383,      692,      387,       64;
  5:    2129,     5132,     4347,     1484,      160;
  6:   12360,    37382,    42732,    22544,     5245,     384;
  7:   75242,   274486,   397002,   286444,   105555,   17730,     896;
  8:  480513,  2056784,  3610779,  3326684,  1701625,  465828,   58471,   2048;
  9: 3216113, 15825164, 32755251, 36874156, 24322615, 9393096, 1984115, 190232, 4608;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+2*n^2-1+(2*n^3+2*n^2-2*n-2)*x+(n^3-n)*x^2)*P(n-2,x) + (n^2+2*n+1+(3*n^2+5*n+2)*x+(3*n^2+4*n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n^2+2*n+1+2*n^2*x)*P(n-1,x) - (n+2*n*x+n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - 1 + (2*n^3 + 2*n^2 - 2*n - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + (n^2 + 2*n + 1 + (3*n^2 + 5*n + 2)*x + (3*n^2 + 4*n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 2*n + 1 + 2*n^2*x)*P[n - 1, x] - (n + 2*n*x + n*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - 1 + (2*n^3 + 2*n^2 - 2*n - 2)*x + (n^3 - n)*x^2)*v[i - 2] + (n^2 + 2*n + 1 + (3*n^2 + 5*n + 2)*x + (3*n^2 + 4*n + 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 2*n + 1 + 2*n^2*x)*v[i - 1] - (n + 2*n*x + n*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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.