Real-rooted atlas

A038314 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(22 + 68\,x + 46\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 2024\,x + 2116\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0484202421164624
1106486679213965697336314432
2234256195923261448648565568447745621381376
35153632538788802253116804711062404925201602059629761453933568
411337990414224024327435285440207286745603250633056027187112832947429689698867482624
52494357888365083290882290067915527980539705601668658302080209340768806414590417219844358176572166722988818432
654875873536917923702784671753255219228091499763712734209652915201228132510330881283956715345927670390767001620047612231936457163239653376
71207269217792227186116439041900102064762889270194922024962907470225544192607925592613785684741143212830727593686859331584396942722192332892219016266905631087100296429568

All 15 rows

Technical details
Node id
n-a37c33d9776b0b38 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2f74b0bddfe64ff1

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:             484,              2024,               2116;
  1:           10648,             66792,             139656,              97336;
  2:          234256,           1959232,            6144864,            8565568,             4477456;
  3:         5153632,          53878880,          225311680,          471106240,           492520160,            205962976;
  4:       113379904,        1422402432,         7435285440,        20728674560,         32506330560,          27187112832,           9474296896;
  5:      2494357888,       36508329088,       229006791552,       798053970560,       1668658302080,        2093407688064,        1459041721984,         435817657216;
  6:     54875873536,      917923702784,      6717532552192,     28091499763712,      73420965291520,      122813251033088,      128395671534592,       76703907670016,       20047612231936;
  7:   1207269217792,    22718611643904,    190010206476288,    927019492202496,    2907470225544192,     6079255926137856,     8474114321283072,     7593686859331584,     3969427221923328,      922190162669056;
  8:  26559922791424,   555343840184320,   5225280678097920,  29134898326364160,  106607241603287040,   267487260750065664,   466076287670568960,   556870369684316160,   436636994411566080,   202881835787192320,    42420747482776576;
  9: 584318301411328, 13439320932460544, 140501991566632960, 881330674372515840, 3685564638285066240, 10788652850252648448, 22558092323255537664, 33690657365901127680, 35222050882532997120, 24548702130250270720, 10265820890831931392, 1951354384207722496;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (22+68*x+46*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 2024*x + 2116*x^2.
%t Mathematica
P[0, x_] := 484 + 2024*x + 2116*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((22 + 68*x + 46*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] = 484 + 2024*x + 2116*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((22 + 68*x + 46*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

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.