Real-rooted atlas

A357613 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(3 + 3\,x - 13\,n - 15\,n x + 14\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} + n + 10\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
26201541
32010516884377
470504126013204953649
5252231079201287010010300336365
692410296450451001001201207425618564369305
7343245045240240680680111384010581485426401162803800305

All 15 rows

Technical details
Node id
n-6dafc159e8998123 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1cc143b67a71a4b2

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:     2,      3;
  2:     6,     20,      15;
  3:    20,    105,     168,       84;
  4:    70,    504,    1260,     1320,      495;
  5:   252,   2310,    7920,    12870,    10010,      3003;
  6:   924,  10296,   45045,   100100,   120120,     74256,     18564;
  7:  3432,  45045,  240240,   680680,  1113840,   1058148,    542640,   116280;
  8: 12870, 194480, 1225224,  4232592,  8817900,  11395440,   8953560,  3922512,   735471;
  9: 48620, 831402, 6046560, 24690120, 62674920, 102965940, 109830336, 73547100, 28120950, 4686825;
%F Formula
The row polynomials satisfy (4*n^2-2*n)*P(n,x) = (14*n^2-13*n+3+(18*n^2-15*n+3)*x)*P(n-1,x) + (n+(10*n-3)*x+(9*n-3)*x^2)*d/dx 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[((14*n^2 - 13*n + 3 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n + (10*n - 3)*x + (9*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 2*n)]];
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] = ((14*n^2 - 13*n + 3 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n + (10*n - 3)*x + (9*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 2*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

Same fingerprint ?

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