Real-rooted atlas

A038336 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(22 + 64\,x + 42\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1848\,x + 1764\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0484184817644096
1106486098411642474088262144
2234256178886451226566519744311169616777216
35153632491937601878307203585859203422865601306912321073741824
411337990412987152646198413760157777804802259091296017251242624548903174468719476736
5249435788833333691776190911143808607444548480115966686528013283456820488453108885762305393332484398046511104
654875873536838104250368560006021836821382048106496510253420723207792961334681674387358194688405749226516489682651996416281474976710656
71207269217792207430801966081584017033195527056075875143682020603546063872385751586066739249095656408494084016917342513152191716509529036840667138384947218014398509481984

All 15 rows

Technical details
Node id
n-c7dce6a64f79d5d9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-16a08a122afbb17c

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,              1848,               1764;
  1:           10648,             60984,             116424,              74088;
  2:          234256,           1788864,            5122656,            6519744,             3111696;
  3:         5153632,          49193760,          187830720,          358585920,           342286560,           130691232;
  4:       113379904,        1298715264,         6198413760,        15777780480,         22590912960,         17251242624,           5489031744;
  5:      2494357888,       33333691776,       190911143808,       607444548480,       1159666865280,       1328345682048,         845310888576,         230539333248;
  6:     54875873536,      838104250368,      5600060218368,     21382048106496,      51025342072320,      77929613346816,       74387358194688,       40574922651648,        9682651996416;
  7:   1207269217792,    20743080196608,    158401703319552,    705607587514368,    2020603546063872,    3857515860667392,     4909565640849408,     4016917342513152,     1917165095290368,      406671383849472;
  8:  26559922791424,   507053071472640,   4356046841287680,  22176238464737280,   74088796689008640,  169730697869365248,   270026110246717440,   294573938450964480,   210888160481940480,    89467704446883840,   17080198121677824;
  9: 584318301411328, 12270684329637888, 117129259510179840, 670831213558302720, 2561355542677155840, 6845804814064398336, 13069263735941124096, 17821723276283351040, 17011644945543198720, 10825592238072944640, 4133407945446033408, 717368321110468608;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (22+64*x+42*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1848*x + 1764*x^2.
%t Mathematica
P[0, x_] := 484 + 1848*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((22 + 64*x + 42*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 + 1848*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((22 + 64*x + 42*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.