Real-rooted atlas

A038314 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(170 + 264\,x + 72\,x^{2} + 60\,n + 72\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 340 + 528\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03405281441012
14600122409504172828072
25800022080029376015206420736745360
3700000348000066240005875200228096024883219208992
4820000050400000125280000158976000105753600328458242985984484441408
59400000068880000021168000003507840000333849600017766604804598415363583180812018269824
61060000000902400000033062400000677376000008418816000064099123200284265676806306398208429981696294334230784
71180000000011448000000048729600000011902464000001828915200000181846425600011537842176004385813299208513637580851597803527133863559680

All 15 rows

Technical details
Node id
n-7ebf4a7fb9dd2d07 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-98c8469350411ee3

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           340,            528,            144;
  1:          4600,          12240,           9504,            1728;
  2:         58000,         220800,         293760,          152064,           20736;
  3:        700000,        3480000,        6624000,         5875200,         2280960,          248832;
  4:       8200000,       50400000,      125280000,       158976000,       105753600,        32845824,         2985984;
  5:      94000000,      688800000,     2116800000,      3507840000,      3338496000,      1776660480,       459841536,        35831808;
  6:    1060000000,     9024000000,    33062400000,     67737600000,     84188160000,     64099123200,     28426567680,      6306398208,       429981696;
  7:   11800000000,   114480000000,   487296000000,   1190246400000,   1828915200000,   1818464256000,   1153784217600,    438581329920,     85136375808,     5159780352;
  8:  130000000000,  1416000000000,  6868800000000,  19491840000000,  35707392000000,  43893964800000,  36369285120000,  19779158016000,   6578719948800,  1135151677440,    61917364224;
  9: 1420000000000, 17160000000000, 93456000000000, 302227200000000, 643230720000000, 942675148800000, 965667225600000, 685820805120000, 326356107264000, 96487892582400, 14984002142208, 743008370688;
%F Formula
The row polynomials satisfy (6*n+11+6*x)*P(n,x) = (60*n+170+(72*n+264)*x+72*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 340 + 528*x + 144*x^2.
%t Mathematica
P[0, x_] := 340 + 528*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 170 + (72*n + 264)*x + 72*x^2)*P[n - 1, x])/(6*n + 11 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 340 + 528*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 170 + (72*n + 264)*x + 72*x^2)*v[i - 1])/(6*n + 11 + 6*x)); 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.