Real-rooted atlas

A048993 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(8\,x + 4\,x^{2} + n - 12\,n x - 28\,n x^{2} - n^{2} + 12\,n^{2} x + 53\,n^{2} x^{2} - 4\,n^{3} x - 38\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-x + 5\,x^{2} + 5\,x^{3} + 4\,n x^{2} - 32\,n x^{3} - 8\,n^{2} x^{2} + 53\,n^{2} x^{3} + 4\,n^{3} x^{2} - 30\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{3} + x^{4} - 6\,n x^{4} + 9\,n^{2} x^{4} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(1 - 4\,x - 3\,n + 12\,n x + 2\,n^{2} - 12\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 4\,n x^{2} + 8\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2145
3112922
41246316104
514022524025531
61605851440775362897
718412605600752522684916787

All 15 rows

Technical details
Node id
n-20a1c4cfd4a13287 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc8b66099215d7ce

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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: 1;
  2: 1,   4;
  3: 1,  12,    9;
  4: 1,  24,   63,    16;
  5: 1,  40,  225,   240,     25;
  6: 1,  60,  585,  1440,    775,     36;
  7: 1,  84, 1260,  5600,   7525,   2268,     49;
  8: 1, 112, 2394, 16800,  42525,  34776,   6223,    64;
  9: 1, 144, 4158, 42336, 173775, 279720, 148225, 16320, 81;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2-n+(4*n^3-12*n^2+12*n-8)*x-(9*n^4-38*n^3+53*n^2-28*n+4)*x^2)*P(n-2,x) - (x-(4*n^3-8*n^2+4*n+5)*x^2-(4*n^4-30*n^3+53*n^2-32*n+5)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3-9*n^2+6*n-1)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-3*n+1+(4*n^3-12*n^2+12*n-4)*x)*P(n-1,x) + (x-(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (4*n^3 - 12*n^2 + 12*n - 8)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*P[n - 2, x] - (x - (4*n^3 - 8*n^2 + 4*n + 5)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 3*n + 1 + (4*n^3 - 12*n^2 + 12*n - 4)*x)*P[n - 1, x] + (x - (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - n + (4*n^3 - 12*n^2 + 12*n - 8)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*v[i - 2] - (x - (4*n^3 - 8*n^2 + 4*n + 5)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 3*n + 1 + (4*n^3 - 12*n^2 + 12*n - 4)*x)*v[i - 1] + (x - (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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. [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.