Real-rooted atlas

A178820 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x + 6\,n^{2} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(24 + 36\,x + 18\,x^{2} + 3\,x^{3} + 8\,n + 48\,n x + 36\,n x^{2} + 7\,n x^{3} + 12\,n^{2} x + 16\,n^{2} x^{2} + 5\,n^{2} x^{3} + 2\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
181624
24016090290
316096010803202520
456044807560448087517955
51792179204032035840140002016111888
6537664512181440215040126000362884116632772
71536021504072576010752008400003628808232076803324240

All 15 rows

Technical details
Node id
n-1a4d5efe81c0009c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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:      8,      16;
  2:     40,     160,      90;
  3:    160,     960,    1080,      320;
  4:    560,    4480,    7560,     4480,      875;
  5:   1792,   17920,   40320,    35840,    14000,     2016;
  6:   5376,   64512,  181440,   215040,   126000,    36288,    4116;
  7:  15360,  215040,  725760,  1075200,   840000,   362880,   82320,    7680;
  8:  42240,  675840, 2661120,  4730880,  4620000,  2661120,  905520,  168960,  13365;
  9: 112640, 2027520, 9123840, 18923520, 22176000, 15966720, 7244160, 2027520, 320760, 22000;
%F Formula
The row polynomials satisfy (4*n+(6*n^2-2*n)*x+n^3*x^2)*P(n,x) = (8*n+24+(12*n^2+48*n+36)*x+(2*n^3+16*n^2+36*n+18)*x^2+(n^3+5*n^2+7*n+3)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 24 + (12*n^2 + 48*n + 36)*x + (2*n^3 + 16*n^2 + 36*n + 18)*x^2 + (n^3 + 5*n^2 + 7*n + 3)*x^3)*P[n - 1, x])/(4*n + (6*n^2 - 2*n)*x + n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 24 + (12*n^2 + 48*n + 36)*x + (2*n^3 + 16*n^2 + 36*n + 18)*x^2 + (n^3 + 5*n^2 + 7*n + 3)*x^3)*v[i - 1])/(4*n + (6*n^2 - 2*n)*x + n^3*x^2)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.