Real-rooted atlas

A104684 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(-8 - 16\,x - 8\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-2}(x) + \left(12 + 6\,x - 8\,n - 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
32220446
4136180728396
58861544936224163606
65944128601048039206403233876
7406361050481076405504014400172864324556
828148884644010452966792802464004838444801283151896
9196893467517609759568770067235828809873921523201126425630915046
10138755445344023688509024821943364673894416656192364492845619227648512305543556
1198365972420390200784906200838713600564076800247121280704390401262592013132806656010243038019876
12700701808329059042468379916008264718000642026880033414796801177102080276629760415641603660800157696204830354866856
1350113719642565110232058717006128792024464006980664240042111083520176946739205164093440102339072013122560099348483686404096304523343996
1435961808432199266393560498192443424741922534496731613625600502153629120245351208960857286144002119163904036036915204000522242635776085196881923065412858696
152588059977521543404868208418454134571268193623214087439843162752572653809728031917575289601300650024960386259793920822684262401217318502411838095366858342419496961638430946859111256
Technical details
Node id
n-ee62a53d652921ca · revision 0 · JSON
Definition source
derived:basis_map_A013609
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 -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:        4,        2;
   3:       22,       20,        4;
   4:      136,      180,       72,        8;
   5:      886,     1544,      936,      224,       16;
   6:     5944,    12860,    10480,     3920,      640,       32;
   7:    40636,   105048,   107640,    55040,    14400,     1728,      64;
   8:   281488,   846440,  1045296,   679280,   246400,    48384,    4480,    128;
   9:  1968934,  6751760,  9759568,  7700672,  3582880,   987392,  152320,  11264,   256;
  10: 13875544, 53440236, 88509024, 82194336, 46738944, 16656192, 3644928, 456192, 27648, 512;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (4*n-8+(8*n-16)*x+(4*n-8)*x^2)*P(n-2,x) - (8*n-12+(4*n-6)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*P[n - 2, x] - (8*n - 12 + (4*n - 6)*x)*P[n - 1, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*v[i - 2] - (8*n - 12 + (4*n - 6)*x)*v[i - 1])/(-n + 1)); 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. [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