Real-rooted atlas

A299989 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 3\,x + 3\,x^{2} + x^{3} + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(9 + 33\,x + 46\,x^{2} + 30\,x^{3} + 9\,x^{4} + x^{5} + 12\,n x + 22\,n x^{2} + 12\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0112
13117122
29579054131224
3272436217154211291912176
4819453348577256303286115623625120480
52433483157953640550070440942561498502455375311188416
67291239368526198018352755418527343924199228815192325544785463711703936
72187430112806659758072137779319995934199132673395154596166213720824347117738574943115204352

All 15 rows

Technical details
Node id
n-7d4fd94324eb099c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-610b16754d4e841f

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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:     1,      1;
  1:     3,     11,       7,        1;
  2:     9,     57,      90,       54,       13,         1;
  3:    27,    243,     621,      715,      421,       129,        19,         1;
  4:    81,    945,    3348,     5772,     5630,      3286,      1156,       236,        25,         1;
  5:   243,   3483,   15795,    36405,    50070,     44094,     25614,      9850,      2455,       375,       31,        1;
  6:   729,  12393,   68526,   198018,   352755,    418527,    343924,    199228,     81519,     23255,     4478,      546,       37,       1;
  7:  2187,  43011,  280665,   975807,  2137779,   3199959,   3419913,   2673395,   1545961,    662137,   208243,    47117,     7385,     749,     43,      1;
  8:  6561, 146529, 1102248,  4480920, 11659788,  21103740,  27841464,  27524712,  20720310,  11968614,  5308888,  1797544,   458220,   85820,  11336,    984,    49,    1;
  9: 19683, 492075, 4192479, 19514601, 58830300, 125000172, 196501788, 235471860, 219151818, 160187338, 92459170, 42163614, 15125772, 4227132, 905004, 144516, 16491, 1251, 55, 1;
%F Formula
The row polynomials satisfy (3+(4*n+3)*x+(2*n+3)*x^2+x^3)*P(n,x) = (9+(12*n+33)*x+(22*n+46)*x^2+(12*n+30)*x^3+(2*n+9)*x^4+x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + x.
%t Mathematica
P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((9 + (12*n + 33)*x + (22*n + 46)*x^2 + (12*n + 30)*x^3 + (2*n + 9)*x^4 + x^5)*P[n - 1, x])/(3 + (4*n + 3)*x + (2*n + 3)*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (12*n + 33)*x + (22*n + 46)*x^2 + (12*n + 30)*x^3 + (2*n + 9)*x^4 + x^5)*v[i - 1])/(3 + (4*n + 3)*x + (2*n + 3)*x^2 + x^3)); 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.