Real-rooted atlas

A056939 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 7\,n + n^{2}\right)\,P_{n}(x) = \left(12 + 27\,n + 10\,n x + 21\,n^{2} + 10\,n^{2} x\right)\,P_{n-1}(x) + \left(48 + 24\,x - 54\,n - 57\,n x - 15\,n x^{2}\right)\,P_{n-1}'(x) + \left(72 + 96\,x + 42\,x^{2} + 6\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2138122
36962161148
44174921942811132
52763403521454954519484
61960934128228507300111068185066
714679329654823956898084211962268981804556
8114683326374002496088124835235173654880431213617939738
99278595239306372596841415355788535029611071201300327734183181128800
107729226122093308027042391518466554076774740198851523149580286740132152401853424464
1165997393320707447862822327640218638282510572140703297212166615457282549505954852167030819201391456
1257561696811966742240429539800378256009719241412352186351594166801257744708201157704202030951174932342983881101327618056
1351137399979188995757535310148183007297367178823184388100039772853345552222316825343569441635673289054663101322175015263748111136518296892
144616910664171835169778848326686169955634335818527922364554236044111912486285637130682989486464354548139222743661500250752102282180402511278624588112954283592578
Technical details
Node id
n-8010313896479d44 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-dd67c79f07eb5aef

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:       3,        1;
  2:      13,        8,        1;
  3:      69,       62,       16,        1;
  4:     417,      492,      194,       28,       1;
  5:    2763,     4035,     2145,      495,      45,       1;
  6:   19609,    34128,    22850,     7300,    1110,      68,      1;
  7:  146793,   296548,   239568,    98084,   21196,    2268,     98,    1;
  8: 1146833,  2637400,  2496088,  1248352,  351736,   54880,   4312,  136,   1;
  9: 9278595, 23930637, 25968414, 15355788, 5350296, 1107120, 130032, 7734, 183, 1;
%F Formula
The row polynomials satisfy (n^2+7*n+12)*P(n,x) = (21*n^2+27*n+12+(10*n^2+10*n)*x)*P(n-1,x) - (54*n-48+(57*n-24)*x+15*n*x^2)*d/dx P(n-1,x) + (72+96*x+42*x^2+6*x^3)*d^2/dx^2 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[((21*n^2 + 27*n + 12 + (10*n^2 + 10*n)*x)*P[n - 1, x] - (54*n - 48 + (57*n - 24)*x + 15*n*x^2)*D[P[n - 1, x], x] + (72 + 96*x + 42*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12)]];
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] = ((21*n^2 + 27*n + 12 + (10*n^2 + 10*n)*x)*v[i - 1] - (54*n - 48 + (57*n - 24)*x + 15*n*x^2)*deriv(v[i - 1], x) + (72 + 96*x + 42*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023