Real-rooted atlas

A108426 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12 + 7\,x + x^{2} - 52\,n - 27\,n x - 2\,n x^{2} + 52\,n^{2} + 26\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(8 + 14\,x + 7\,x^{2} + x^{3} - 20\,n - 32\,n x - 13\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2109221
36685365192
4498833516140142001
540668361680027305404222539
634970853058583045540134202079132267276
731206688068510554606956952720906306380084293287496

All 15 rows

Technical details
Node id
n-c9014249fe74edd2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b24c47ab26a256b2

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,        1;
  2:       10,        9,         2;
  3:       66,       85,        36,         5;
  4:      498,      833,       516,       140,       14;
  5:     4066,     8361,      6800,      2730,      540,       42;
  6:    34970,    85305,     85830,     45540,    13420,     2079,     132;
  7:   312066,   880685,   1055460,    695695,   272090,    63063,    8008,     429;
  8:  2862562,  9173505,  12754952,  10042760,  4893420,  1509508,  287560,   30888,   1430;
  9: 26824386, 96220561, 152235072, 139351380, 81276048, 31298904, 7951104, 1283568, 119340, 4862;
%F Formula
The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (52*n^2-52*n+12+(26*n^2-27*n+7)*x+(n^2-2*n+1)*x^2)*P(n-1,x) - (20*n-8+(32*n-14)*x+(13*n-7)*x^2+(n-1)*x^3)*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[((52*n^2 - 52*n + 12 + (26*n^2 - 27*n + 7)*x + (n^2 - 2*n + 1)*x^2)*P[n - 1, x] - (20*n - 8 + (32*n - 14)*x + (13*n - 7)*x^2 + (n - 1)*x^3)*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] = ((52*n^2 - 52*n + 12 + (26*n^2 - 27*n + 7)*x + (n^2 - 2*n + 1)*x^2)*v[i - 1] - (20*n - 8 + (32*n - 14)*x + (13*n - 7)*x^2 + (n - 1)*x^3)*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. [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.