Real-rooted atlas

A104684 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(-50 - 240\,x - 288\,x^{2} + 25\,n + 120\,n x + 144\,n x^{2}\right)\,P_{n-2}(x) + \left(21 + 36\,x - 14\,n - 24\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
271219
361192144397
45952700388817288911
5614536480777606912020736210241
6655274812601362240183168011404802488325130019
77129096243264220644003988224038257920179159042985984128062621
87863667799974843391930087714224001005177600736791552271724544358318083248002063

All 15 rows

Technical details
Node id
n-36ff311318433b63 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2999dcf3b57aaeda

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:         7,          12;
   3:        61,         192,         144;
   4:       595,        2700,        3888,         1728;
   5:      6145,       36480,       77760,        69120,        20736;
   6:     65527,      481260,     1362240,      1831680,      1140480,       248832;
   7:    712909,     6243264,    22064400,     39882240,     38257920,     17915904,      2985984;
   8:   7863667,    79997484,   339193008,    771422400,   1005177600,    736791552,    271724544,     35831808;
   9:  87615745,  1015472640,  5023823616,  13780924416,  22745318400,  22852730880,  13377208320,   4013162496,   429981696;
  10: 983726695, 12796248780, 72366255360, 232524794880, 465072233472, 595390279680, 481579499520, 232190115840, 58047528960, 5159780352;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (25*n-50+(120*n-240)*x+(144*n-288)*x^2)*P(n-2,x) - (14*n-21+(24*n-36)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 7 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((25*n - 50 + (120*n - 240)*x + (144*n - 288)*x^2)*P[n - 2, x] - (14*n - 21 + (24*n - 36)*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] = 7 + 12*x; for(n = 3, N, my(i = n); v[i] = ((25*n - 50 + (120*n - 240)*x + (144*n - 288)*x^2)*v[i - 2] - (14*n - 21 + (24*n - 36)*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