Real-rooted atlas

A008459 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-6 - 4\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22220446
3136180728396
48861544936224163606
55944128601048039206403233876
6406361050481076405504014400172864324556
728148884644010452966792802464004838444801283151896
8196893467517609759568770067235828809873921523201126425630915046
9138755445344023688509024821943364673894416656192364492845619227648512305543556
1098365972420390200784906200838713600564076800247121280704390401262592013132806656010243038019876
11700701808329059042468379916008264718000642026880033414796801177102080276629760415641603660800157696204830354866856
1250113719642565110232058717006128792024464006980664240042111083520176946739205164093440102339072013122560099348483686404096304523343996
1335961808432199266393560498192443424741922534496731613625600502153629120245351208960857286144002119163904036036915204000522242635776085196881923065412858696
142588059977521543404868208418454134571268193623214087439843162752572653809728031917575289601300650024960386259793920822684262401217318502411838095366858342419496961638430946859111256

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-a3fd45efabb10cb9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-011323e4bbdda656)
Fingerprint
f6-50f668b4dcf4d56e

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:        4,        2;
  2:       22,       20,        4;
  3:      136,      180,       72,        8;
  4:      886,     1544,      936,      224,       16;
  5:     5944,    12860,    10480,     3920,      640,       32;
  6:    40636,   105048,   107640,    55040,    14400,     1728,      64;
  7:   281488,   846440,  1045296,   679280,   246400,    48384,    4480,    128;
  8:  1968934,  6751760,  9759568,  7700672,  3582880,   987392,  152320,  11264,   256;
  9: 13875544, 53440236, 88509024, 82194336, 46738944, 16656192, 3644928, 456192, 27648, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (10*n-6+(6*n-4)*x)*P(n-1,x) - (6+10*x+4*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[((10*n - 6 + (6*n - 4)*x)*P[n - 1, x] - (6 + 10*x + 4*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] = ((10*n - 6 + (6*n - 4)*x)*v[i - 1] - (6 + 10*x + 4*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