Real-rooted atlas

A008306 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-9\,n x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-4\,n x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(4\,n x - 4\,n^{3} x\right)\,P_{n-3}(x) + \left(-4\,n x^{2} + 4\,n^{3} x^{2}\right)\,P_{n-3}'(x) + \left(n + 4\,n x + 4\,n^{2} x - n^{3}\right)\,P_{n-2}(x) + \left(-4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-1 + n + 2\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 6 + 12\,x\)
\(P_{3}(x) = 24 + 80\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
261218
32480104
4120520135775
5720369618906306
650402923221420168057372
74032025689623788840320575424
836288024935042732940705600236256318549
9362880026547840331095601128512086625075437570
103991680030799296042591938417758048021656250374220973440094
114790016003870097920582782428828373030404714710001945944013505157288
12622702080052386946560847705098244672840972897362265006583777206621615200514112747
13871782912007602413414401308695222016799245807360198373175000187221434404635130503172919493506
14130767436800011777241369600213984877766401424923403904040707947280004911346440002039457420012972960053315091229080
Technical details
Node id
n-f545432d58040336 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a32349ec41bc229b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:       2;
  2:       6,       12;
  3:      24,       80;
  4:     120,      520,      135;
  5:     720,     3696,     1890;
  6:    5040,    29232,    21420,     1680;
  7:   40320,   256896,   237888,    40320;
  8:  362880,  2493504,  2732940,   705600,  23625;
  9: 3628800, 26547840, 33109560, 11285120, 866250;
%F Formula
The row polynomials satisfy n*P(n,x) = (9*n^3-9*n)*x^2*P(n-4,x) + (4*n^3-4*n)*x^3*d/dx P(n-4,x) - (4*n^3-4*n)*x*P(n-3,x) + (4*n^3-4*n)*x^2*d/dx P(n-3,x) - (n^3-n-(4*n^2+4*n)*x)*P(n-2,x) - (4*n^2+4*n)*x^2*d/dx P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 6 + 12*x, P(3,x) = 24 + 80*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 6 + 12*x;
P[3, x_] := 24 + 80*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 9*n)*x^2*P[n - 4, x] + (4*n^3 - 4*n)*x^3*D[P[n - 4, x], x] - (4*n^3 - 4*n)*x*P[n - 3, x] + (4*n^3 - 4*n)*x^2*D[P[n - 3, x], x] - (n^3 - n - (4*n^2 + 4*n)*x)*P[n - 2, x] - (4*n^2 + 4*n)*x^2*D[P[n - 2, x], x] + (2*n^2 + n - 1)*P[n - 1, 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; v[2] = 2; v[3] = 6 + 12*x; v[4] = 24 + 80*x; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 - 9*n)*x^2*v[i - 4] + (4*n^3 - 4*n)*x^3*deriv(v[i - 4], x) - (4*n^3 - 4*n)*x*v[i - 3] + (4*n^3 - 4*n)*x^2*deriv(v[i - 3], x) - (n^3 - n - (4*n^2 + 4*n)*x)*v[i - 2] - (4*n^2 + 4*n)*x^2*deriv(v[i - 2], x) + (2*n^2 + n - 1)*v[i - 1])/(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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.