Real-rooted atlas

A204420 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 36\,x - 12\,x^{2} + 108\,n - 120\,n x + 32\,n x^{2} - 144\,n^{2} + 112\,n^{2} x - 16\,n^{2} x^{2} + 80\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 4\,x - 12\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2242448
369010803602130
43696072240403206720156240
5319221071820005216400151200015120017253810
640590396010094792408681904003359664005987520039916802683406880
77140722589019170899748018609386796086875588800210378168002542700160121080960559787278050

All 15 rows

Technical details
Node id
n-351bdce99219d68c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8ede4fc391b0c029

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:                0;
  1:                2;
  2:               24,                24;
  3:              690,              1080,               360;
  4:            36960,             72240,             40320,              6720;
  5:          3192210,           7182000,           5216400,           1512000,           151200;
  6:        405903960,        1009479240,         868190400,         335966400,         59875200,         3991680;
  7:      71407225890,      191708997480,      186093867960,       86875588800,      21037816800,      2542700160,      121080960;
  8:   16614469872000,    47418957986400,    50498025177600,    26990762198400,    7991343360000,   1329468940800,   116237721600,    4151347200;
  9: 4941282024929250, 14833862809365600, 17012296133632800, 10089574083388800, 3456605624472000, 710263332979200, 86328302860800, 5716405094400, 158789030400;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(32*n^3-112*n^2+120*n-36)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (8*n - 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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (8*n - 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.