Real-rooted atlas

A204420 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 54\,x - 27\,x^{2} + 108\,n - 180\,n x + 72\,n x^{2} - 144\,n^{2} + 168\,n^{2} x - 36\,n^{2} x^{2} + 80\,n^{3} - 48\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 6\,x - 12\,n + 12\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
2459027162
31260346524304057560
466465217980217350793808505589680
55681340209478152568510013522950306180022963569128640
6716911965288018423040658067452677544100879884775136403190757795511364312960
71253723070605384560831658437394265306491632697552697078384006091412827568959390502955402452494544532480

All 15 rows

Technical details
Node id
n-a58c3ee71fbac741 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f681b7570d759557

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:                4,                 3;
  2:               45,                90,                27;
  3:             1260,              3465,              2430,               405;
  4:            66465,            217980,            217350,             79380,              8505;
  5:          5681340,          20947815,          25685100,          13522950,           3061800,            229635;
  6:        716911965,        2880184230,        4065806745,        2677544100,         879884775,         136403190,          7577955;
  7:     125372307060,      538456083165,      843739426530,      649163269755,      269707838400,       60914128275,       6895939050,       295540245;
  8:   29030535158625,   131614360277400,   223777162709100,   193451011954200,    94844447047350,    27280334881800,    4509944138700,    390113123400,    13299311025;
  9: 8599375008724500, 40788114358466475, 74104595111547000, 70217301947492700, 39018216160123200, 13340063069383050, 2823390533363400, 357219494131500, 24417535041900, 678264862275;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 6)*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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 6)*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.