Real-rooted atlas

A290596 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 10\,x + 42\,x^{2} + 6\,n x - 39\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x^{2} + 48\,x^{3} - 18\,n x^{3}\right)\,P_{n-1}'(x) + \left(9\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
31101021
412412080225
514452817608803213
6170154012320308001232057051
711023570523603141606283202094401207953
8114071401666001832600879648014660800418880029652561
911841288043792076636006743968026975872038536960096342400827024985
101234215281004640256183203586564802630147520901764864011272060800250490240025810060563
1112903393020810407283640014858625601733506320010896325440032688976320036321084800072642169600890601912621
12135251040398112018313152051276825608717060352087170603520047943831936001278502184960012785021849600232454942720033653167805313
13142073920714560041801760015383047680358937779200523023621120045764566848000223737882368000536970917683200488155379712000813592299520001381592958838821
14149410374012172160882481600413001388801266537592320253307518464003229670860416002511966224768000110526513889792002411487575777280020095729798144000309165073817600061216480478217195
151574141778198489201746704960101308887680395104661952010385608257024018174814449792002059812304309760014418686130168320057674744520673280011534948904134656008873037618565120001267576802652160002911014153736460073
Technical details
Node id
n-b3547558d8fdf454 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-94f770baffeadbcb

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,  10,    10;
   4: 1,  24,   120,      80;
   5: 1,  44,   528,    1760,      880;
   6: 1,  70,  1540,   12320,    30800,     12320;
   7: 1, 102,  3570,   52360,   314160,    628320,     209440;
   8: 1, 140,  7140,  166600,  1832600,   8796480,   14660800,    4188800;
   9: 1, 184, 12880,  437920,  7663600,  67439680,  269758720,  385369600,    96342400;
  10: 1, 234, 21528, 1004640, 25618320, 358656480, 2630147520, 9017648640, 11272060800, 2504902400;
%F Formula
The row polynomials satisfy P(n,x) = (1+(6*n-10)*x+(9*n^2-39*n+42)*x^2)*P(n-1,x) - (6*x^2+(18*n-48)*x^3)*d/dx P(n-1,x) + 9*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 10)*x + (9*n^2 - 39*n + 42)*x^2)*P[n - 1, x] - (6*x^2 + (18*n - 48)*x^3)*D[P[n - 1, x], x] + 9*x^4*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (6*n - 10)*x + (9*n^2 - 39*n + 42)*x^2)*v[i - 1] - (6*x^2 + (18*n - 48)*x^3)*deriv(v[i - 1], x) + 9*x^4*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.