Real-rooted atlas

A132393 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-256 + 256\,x - 96\,x^{2} + 16\,x^{3} - x^{4} + 256\,n - 192\,n x + 48\,n x^{2} - 4\,n x^{3} - 96\,n^{2} + 48\,n^{2} x - 6\,n^{2} x^{2} + 16\,n^{3} - 4\,n^{3} x - n^{4}\right)\,P_{n-4}(x) + \left(-175 + 148\,x - 42\,x^{2} + 4\,x^{3} + 148\,n - 84\,n x + 12\,n x^{2} - 42\,n^{2} + 12\,n^{2} x + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-55 + 36\,x - 6\,x^{2} + 36\,n - 12\,n x - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-10 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8\,x\)
\(P_{2}(x) = 8\,x + 27\,x^{2}\)
\(P_{3}(x) = 16\,x + 81\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1088
2082735
30168164161
4048297384125854
501921350224012502165248
6096073981440010625324034336966
7057604762810393691875378007203512294714
8040320352836840448846125423360110446143367292628600
903225602958768755993684105004848984155584827955226244100025963392
100290304027717552750528009046000058174200217026394838400634230450001331281529192
110290304002869732808162288641051187500738056880309404865807797761323135013200007320517283326287848
1203193344003254683680965875046413157259500993507768045754605051350429696260561367326700002562175114048219742546905712
130383201280040133957760123619811328176751861000141956676576706820614502303499827250495161867494630007422987046949761713662744585889457328
14049816166400534674494080170218989158425392191370002150864011296114428874359840496289433698441588817166696530001962523563157405248819700724970433758643254959008
Technical details
Node id
n-d0a6f2af472048c1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d9c23e462c1c1f8c

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,      8;
  2: 0,      8,      27;
  3: 0,     16,      81,      64;
  4: 0,     48,     297,     384,     125;
  5: 0,    192,    1350,    2240,    1250,     216;
  6: 0,    960,    7398,   14400,   10625,    3240,     343;
  7: 0,   5760,   47628,  103936,   91875,   37800,    7203,    512;
  8: 0,  40320,  352836,  840448,  846125,  423360,  110446,  14336,   729;
  9: 0, 322560, 2958768, 7559936, 8410500, 4848984, 1555848, 279552, 26244, 1000;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256+(4*n^3-48*n^2+192*n-256)*x+(6*n^2-48*n+96)*x^2+(4*n-16)*x^3+x^4)*P(n-4,x) + (4*n^3-42*n^2+148*n-175+(12*n^2-84*n+148)*x+(12*n-42)*x^2+4*x^3)*P(n-3,x) - (6*n^2-36*n+55+(12*n-36)*x+6*x^2)*P(n-2,x) + (4*n-10+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8*x, P(2,x) = 8*x + 27*x^2, P(3,x) = 16*x + 81*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8*x;
P[2, x_] := 8*x + 27*x^2;
P[3, x_] := 16*x + 81*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*P[n - 2, x] + (4*n - 10 + 4*x)*P[n - 1, x]];
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] = 8*x; v[3] = 8*x + 27*x^2; v[4] = 16*x + 81*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*v[i - 2] + (4*n - 10 + 4*x)*v[i - 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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.