Real-rooted atlas

A133214 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-25 - 120\,x - 144\,x^{2} - 25\,n - 120\,n x - 144\,n x^{2}\right)\,P_{n-2}(x) + \left(21 + 36\,x + 14\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 61 + 192\,x + 144\,x^{2}\)
\(P_{1}(x) = 595 + 2700\,x + 3888\,x^{2} + 1728\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
061192144397
15952700388817288911
2614536480777606912020736210241
3655274812601362240183168011404802488325130019
47129096243264220644003988224038257920179159042985984128062621
57863667799974843391930087714224001005177600736791552271724544358318083248002063
6876157451015472640502382361613780924416227453184002285273088013377208320401316249642998169683326238209
798372669512796248780723662553602325247948804650722334725953902796804815794995202321901158405804752896051597803522156110463539

All 15 rows

Technical details
Node id
n-f018f7f0d15d5041 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2999dcf3b57aaeda

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           61,           192,            144;
  1:          595,          2700,           3888,           1728;
  2:         6145,         36480,          77760,          69120,           20736;
  3:        65527,        481260,        1362240,        1831680,         1140480,          248832;
  4:       712909,       6243264,       22064400,       39882240,        38257920,        17915904,         2985984;
  5:      7863667,      79997484,      339193008,      771422400,      1005177600,       736791552,       271724544,        35831808;
  6:     87615745,    1015472640,     5023823616,    13780924416,     22745318400,     22852730880,     13377208320,      4013162496,       429981696;
  7:    983726695,   12796248780,    72366255360,   232524794880,    465072233472,    595390279680,    481579499520,    232190115840,     58047528960,     5159780352;
  8:  11112210781,  160308442560,  1020163591440,  3758281390080,   8835828986880,  13765649006592,  14263030333440,   9571392552960,   3889184440320,   825564856320,    61917364224;
  9: 126142119187, 1998734904012, 14136195150000, 58741025999040, 158768050129920, 291673577631744, 369185616543744, 318906674380800, 181553321410560, 63284706017280, 11578547109888, 743008370688;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(25*n+25+(120*n+120)*x+(144*n+144)*x^2)*P(n-2,x) + (14*n+21+(24*n+36)*x)*P(n-1,x) for n >= 2, with P(0,x) = 61 + 192*x + 144*x^2, P(1,x) = 595 + 2700*x + 3888*x^2 + 1728*x^3.
%t Mathematica
P[0, x_] := 61 + 192*x + 144*x^2;
P[1, x_] := 595 + 2700*x + 3888*x^2 + 1728*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(25*n + 25 + (120*n + 120)*x + (144*n + 144)*x^2)*P[n - 2, x] + (14*n + 21 + (24*n + 36)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 61 + 192*x + 144*x^2; v[2] = 595 + 2700*x + 3888*x^2 + 1728*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(25*n + 25 + (120*n + 120)*x + (144*n + 144)*x^2)*v[i - 2] + (14*n + 21 + (24*n + 36)*x)*v[i - 1])/(n + 2)); 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