Real-rooted atlas

A056939 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 5\,n + 6\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(24 + 26\,n - 25\,n x + 9\,n^{2} + 15\,n^{2} x + n^{3} + 10\,n^{3} x\right)\,P_{n-1}(x) + \left(64\,x + 36\,x^{2} + 33\,n x - 21\,n x^{2} + 3\,n^{2} x - 15\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(14\,x^{2} + 14\,x^{3} + 6\,n x^{2} + 6\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
28614
320601292
44030024020600
57010502100700303950
6112294011760980016804226334
71687056493928232035280352856177800
8240151201693444939204233601058406720721214616
93302970049896023284803492720174636027720011880908385720
104405445013068009147600219542401920996060984006534001980011058445200
11572943803114540311454001121234401569728168720712018687240141570031460132410792800
127281561566870864944743804858682401020323304906954048340107768515314802862860480481562909198032
13910248430142101962605202601842250410552675123073690016404421400984117234117013026013054654607098018220742521982
1411203822002782416066314248062524862402579150574049126677600442140098401875745872036472836403060657609937200101920210148796876830
Technical details
Node id
n-ce334d10a4ef74db · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f1adc5b977bebb3c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   2;
  2:   8,     6;
  3:  20,    60,     12;
  4:  40,   300,    240,      20;
  5:  70,  1050,   2100,     700,      30;
  6: 112,  2940,  11760,    9800,    1680,      42;
  7: 168,  7056,  49392,   82320,   35280,    3528,     56;
  8: 240, 15120, 169344,  493920,  423360,  105840,   6720,    72;
  9: 330, 29700, 498960, 2328480, 3492720, 1746360, 277200, 11880, 90;
%F Formula
The row polynomials satisfy (n^3+6*n^2+5*n-12)*P(n,x) = (n^3+9*n^2+26*n+24+(10*n^3+15*n^2-25*n)*x)*P(n-1,x) + ((3*n^2+33*n+64)*x-(15*n^2+21*n-36)*x^2)*d/dx P(n-1,x) + ((6*n+14)*x^2+(6*n+14)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 + 15*n^2 - 25*n)*x)*P[n - 1, x] + ((3*n^2 + 33*n + 64)*x - (15*n^2 + 21*n - 36)*x^2)*D[P[n - 1, x], x] + ((6*n + 14)*x^2 + (6*n + 14)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 + 15*n^2 - 25*n)*x)*v[i - 1] + ((3*n^2 + 33*n + 64)*x - (15*n^2 + 21*n - 36)*x^2)*deriv(v[i - 1], x) + ((6*n + 14)*x^2 + (6*n + 14)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^3 + 6*n^2 + 5*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