Real-rooted atlas

A038281 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(22 + 6\,x + 11\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 66 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0661884
11089594811764
21597213068356432432928
321961523958098010178201215576240
42898918395307021562205880608019043749680832
537202781608772784150723515093540308731533679815309158120256
646769210489286674473052733633205788090561240148191121347192524882529924096
7578768978712627686808120537010446574746024224139069048903069666686004519631217714739846304512

All 15 rows

Technical details
Node id
n-bfe674a999ae5db6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4f0d71d8efeb5cf0

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           66,            18;
  1:         1089,           594,            81;
  2:        15972,         13068,          3564,           324;
  3:       219615,        239580,         98010,         17820,         1215;
  4:      2898918,       3953070,       2156220,        588060,        80190,         4374;
  5:     37202781,      60877278,      41507235,      15093540,      3087315,       336798,       15309;
  6:    467692104,     892866744,     730527336,     332057880,     90561240,     14819112,     1347192,       52488;
  7:   5787689787,   12627686808,   12053701044,    6574746024,   2241390690,    489030696,    66686004,     5196312,     177147;
  8:  70738430730,  173630693610,  189415302120,  120537010440,  49310595180,  13448344140,  2445153480,   285797160,   19486170,   590490;
  9: 855935011833, 2334368214090, 2864906444565, 2083568323320, 994430336130, 325449928188, 73965892770, 11527152120, 1178913285, 71449290, 1948617;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(3*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 66 + 18*x.
%t Mathematica
P[0, x_] := 66 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (3*n + 6)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 66 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (3*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.