Real-rooted atlas

A008459 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2412622
38486020136
41616036028070886
5324801680224012602525944
664134467201344012600554492440636
7128358424192672009240066528240243432281488
8256921680640295680554400576576336336102960128701968934
951223040253440118272028828804036032336336016473604375804862013875544
101024563207603204392960134534402421619226906880186700807876440184756018475698365972
1120481351682196480153753605765760012915302418296678416803072099768240369512007759752705432700701808
124096319488615014451251200230630400627314688109780070412770334729976824005173168001707145443244987227041565011371964
138192745472167731201640038408712704002822916096595948953685135564808380532160569048480026176230087787969281352078001040060035961808432
1416384172032044728320506920960313657344011918979072297974476805108133888061457235840523524601603141147609612979948800351540280056163240040116600258805997752
Technical details
Node id
n-75813ed3ff97a62c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a6dfa0000f2b0fe9

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,     2;
  2:   4,    12,      6;
  3:   8,    48,     60,      20;
  4:  16,   160,    360,     280,      70;
  5:  32,   480,   1680,    2240,    1260,     252;
  6:  64,  1344,   6720,   13440,   12600,    5544,     924;
  7: 128,  3584,  24192,   67200,   92400,   66528,   24024,    3432;
  8: 256,  9216,  80640,  295680,  554400,  576576,  336336,  102960,  12870;
  9: 512, 23040, 253440, 1182720, 2882880, 4036032, 3363360, 1647360, 437580, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+2*n*x)*P(n-1,x) + (4*x+2*x^2)*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[((2*n + 2*n*x)*P[n - 1, x] + (4*x + 2*x^2)*D[P[n - 1, x], x])/(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] = ((2*n + 2*n*x)*v[i - 1] + (4*x + 2*x^2)*deriv(v[i - 1], x))/(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. [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