Real-rooted atlas

A038331 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 44\,x + 27\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 918\,x + 729\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02899187291936
1491323409371791968385184
283521530604126408613384445314413748096
314198571127533535815770568838704517248514348907164916224
4241375692300168349133021351934051580230379673514635885143874204897256313856
54103386734562000541217365908135753803450591383937155870835165834610303819110460353203319277809664
66975757441886331533684926960584281565034538536310705386327039477860850963135006596988142260803560828242953648114048223625216
71185878764971695109058163107689281342123990838073266895075848216062151002817754922159885336446388108829514724012432117190815937625597484987618121839509504

All 15 rows

Technical details
Node id
n-e09f58916fb51b22 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-df32274ca8b60243

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:            289,             918,              729;
  1:           4913,           23409,            37179,             19683;
  2:          83521,          530604,          1264086,           1338444,            531441;
  3:        1419857,        11275335,         35815770,          56883870,          45172485,           14348907;
  4:       24137569,       230016834,        913302135,        1934051580,        2303796735,         1463588514,          387420489;
  5:      410338673,      4562000541,      21736590813,       57538034505,       91383937155,        87083516583,        46103038191,        10460353203;
  6:     6975757441,     88633153368,     492696058428,     1565034538536,     3107053863270,      3947786085096,      3135006596988,      1422608035608,       282429536481;
  7:   118587876497,   1695109058163,   10768928134212,    39908380732668,    95075848216062,    151002817754922,    159885336446388,    108829514724012,     43211719081593,      7625597484987;
  8:  2015993900449,  32018726654190,  228839722852005,   969203532079080,  2693815699455090,   5134095803667348,   6795126798971490,   6167005834360680,   3672996121935405,   1296351572447790,   205891132094649;
  9: 34271896307633, 598750188433353, 4754780908147215, 22655132562348495, 71963362256871690, 160012652547632346, 254137742281533726, 288307522756361790, 228950091600640245, 121208872023868365, 38501641701699363, 5559060566555523;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+44*x+27*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 918*x + 729*x^2.
%t Mathematica
P[0, x_] := 289 + 918*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 44*x + 27*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 918*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 44*x + 27*x^2)*v[i - 1])/(1 + 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

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.