Real-rooted atlas

A048993 → (θ+1)² → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(3\,x^{2} + 29\,x^{3} + 87\,x^{4} + 117\,x^{5} + 74\,x^{6} + 18\,x^{7} + 5\,n x^{2} + n x^{3} - 42\,n x^{4} - 82\,n x^{5} - 59\,n x^{6} - 15\,n x^{7} - n^{2} x^{2} - n^{2} x^{3} + 6\,n^{2} x^{4} + 14\,n^{2} x^{5} + 11\,n^{2} x^{6} + 3\,n^{2} x^{7}\right)\,P_{n-2}(x) + \left(-6\,x^{2} - 8\,x^{3} + 46\,x^{4} + 144\,x^{5} + 166\,x^{6} + 88\,x^{7} + 18\,x^{8} + 2\,n x^{2} + 4\,n x^{3} - 10\,n x^{4} - 40\,n x^{5} - 50\,n x^{6} - 28\,n x^{7} - 6\,n x^{8}\right)\,P_{n-2}'(x) + \left(-x^{2} - 3\,x^{3} + 3\,x^{4} + 25\,x^{5} + 45\,x^{6} + 39\,x^{7} + 17\,x^{8} + 3\,x^{9}\right)\,P_{n-2}''(x) + \left(-1 + 2\,x + 10\,x^{2} + 8\,x^{3} - x^{4} - 2\,x^{5} - 2\,n x - 4\,n x^{2} + 4\,n x^{4} + 2\,n x^{5}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 4\,x^{3} - 4\,x^{4} - 6\,x^{5} - 2\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1044
2041317
304354786
40475234188501
50415182914718253280
604299259676599246393723741
704591771133764659165948220270187738

All 15 rows

Technical details
Node id
n-9b0bbaccdbcc9299 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9d587861d4436aa8

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 4;
  2: 0, 4,   13;
  3: 0, 4,   35,    47;
  4: 0, 4,   75,   234,    188;
  5: 0, 4,  151,   829,   1471,     825;
  6: 0, 4,  299,  2596,   7659,    9246,    3937;
  7: 0, 4,  591,  7711,  33764,   65916,   59482,   20270;
  8: 0, 4, 1171, 22398, 137090,  385460,  553373,  395526,  111835;
  9: 0, 4, 2327, 64577, 533069, 2028091, 4121647, 4636753, 2729705, 657423;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -((n^2-5*n-3)*x^2+(n^2-n-29)*x^3-(6*n^2-42*n+87)*x^4-(14*n^2-82*n+117)*x^5-(11*n^2-59*n+74)*x^6-(3*n^2-15*n+18)*x^7)*P(n-2,x) + ((2*n-6)*x^2+(4*n-8)*x^3-(10*n-46)*x^4-(40*n-144)*x^5-(50*n-166)*x^6-(28*n-88)*x^7-(6*n-18)*x^8)*d/dx P(n-2,x) - (x^2+3*x^3-3*x^4-25*x^5-45*x^6-39*x^7-17*x^8-3*x^9)*d^2/dx^2 P(n-2,x) - (1+(2*n-2)*x+(4*n-10)*x^2-8*x^3-(4*n-1)*x^4-(2*n-2)*x^5)*P(n-1,x) + (2*x+6*x^2+4*x^3-4*x^4-6*x^5-2*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^2 - 5*n - 3)*x^2 + (n^2 - n - 29)*x^3 - (6*n^2 - 42*n + 87)*x^4 - (14*n^2 - 82*n + 117)*x^5 - (11*n^2 - 59*n + 74)*x^6 - (3*n^2 - 15*n + 18)*x^7)*P[n - 2, x] + ((2*n - 6)*x^2 + (4*n - 8)*x^3 - (10*n - 46)*x^4 - (40*n - 144)*x^5 - (50*n - 166)*x^6 - (28*n - 88)*x^7 - (6*n - 18)*x^8)*D[P[n - 2, x], x] - (x^2 + 3*x^3 - 3*x^4 - 25*x^5 - 45*x^6 - 39*x^7 - 17*x^8 - 3*x^9)*D[P[n - 2, x], {x, 2}] - (1 + (2*n - 2)*x + (4*n - 10)*x^2 - 8*x^3 - (4*n - 1)*x^4 - (2*n - 2)*x^5)*P[n - 1, x] + (2*x + 6*x^2 + 4*x^3 - 4*x^4 - 6*x^5 - 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n^2 - 5*n - 3)*x^2 + (n^2 - n - 29)*x^3 - (6*n^2 - 42*n + 87)*x^4 - (14*n^2 - 82*n + 117)*x^5 - (11*n^2 - 59*n + 74)*x^6 - (3*n^2 - 15*n + 18)*x^7)*v[i - 2] + ((2*n - 6)*x^2 + (4*n - 8)*x^3 - (10*n - 46)*x^4 - (40*n - 144)*x^5 - (50*n - 166)*x^6 - (28*n - 88)*x^7 - (6*n - 18)*x^8)*deriv(v[i - 2], x) - (x^2 + 3*x^3 - 3*x^4 - 25*x^5 - 45*x^6 - 39*x^7 - 17*x^8 - 3*x^9)*deriv(deriv(v[i - 2], x), x) - (1 + (2*n - 2)*x + (4*n - 10)*x^2 - 8*x^3 - (4*n - 1)*x^4 - (2*n - 2)*x^5)*v[i - 1] + (2*x + 6*x^2 + 4*x^3 - 4*x^4 - 6*x^5 - 2*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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.