Real-rooted atlas

A038336 → x ↦ x/(x+1) → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(34 + 44\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 2992\,x + 1936\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01156299219366084
13930415259219747285184474552
2133633669175041342809611585024374809637015056
3454354242939939207609254409847270406371763201649162242887174368
4154480441611994951936388071974406696143872064991984640336429096967256313856225199600704
5525233501444757997601281847222598144398420560384051560307814404003506253824172700269772831927780966417565568854912
61785793904896184882192506888374075778252821674078484889635061009313792036298456701337623487236689100886843564228608140482236252161370114370683136
760716992766464707174386338816366066741163622411053780027293696214573377000407042776831937652326423956981422882816132870653269770244298756429316096618121839509504106868920913284608

All 15 rows

Technical details
Node id
n-4113fe25dfc072d5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-6f86be61a76d4bf4

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:              1156,               2992,                1936;
  1:             39304,             152592,              197472,                85184;
  2:           1336336,            6917504,            13428096,             11585024,              3748096;
  3:          45435424,          293993920,           760925440,            984727040,            637176320,             164916224;
  4:        1544804416,        11994951936,         38807197440,          66961438720,          64991984640,           33642909696,            7256313856;
  5:       52523350144,       475799760128,       1847222598144,        3984205603840,        5156030781440,         4003506253824,         1727002697728,          319277809664;
  6:     1785793904896,     18488219250688,      83740757782528,      216740784848896,      350610093137920,       362984567013376,       234872366891008,        86843564228608,       14048223625216;
  7:    60716992766464,    707174386338816,    3660667411636224,    11053780027293696,    21457337700040704,     27768319376523264,     23956981422882816,     13287065326977024,     4298756429316096,      618121839509504;
  8:  2064377754059776,  26715476817244160,  155578364994539520,   536897887039979520,  1215915803002306560,   1888245717603581952,   2036343420945039360,   1505867403724062720,   730788592983736320,   210161425433231360,    27197360938418176;
  9: 70188843638032384, 999158832964931584, 6465145389773086720, 25099976219119042560, 64964644331837521920, 117700649730623275008, 152318487886688944128, 140798602248199864320, 91104977925305794560, 39300186556014264320, 10171812990968397824, 1196683881290399744;
%F Formula
The row polynomials satisfy P(n,x) = (34+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 2992*x + 1936*x^2.
%t Mathematica
P[0, x_] := 1156 + 2992*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 44*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1156 + 2992*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 44*x)*v[i - 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.