Real-rooted atlas

A038301 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0176418484844096
1740881164246098410648262144
2311169665197445122656178886423425616777216
31306912323422865603585859201878307204919376051536321073741824
454890317441725124262422590912960157777804806198413760129871526411337990468719476736
5230539333248845310888576132834568204811596668652806074445484801909111438083333369177624943578884398046511104
696826519964164057492265164874387358194688779296133468165102534207232021382048106496560006021836883810425036854875873536281474976710656
74066713838494721917165095290368401691734251315249095656408494083857515860667392202060354606387270560758751436815840170331955220743080196608120726921779218014398509481984

All 15 rows

Technical details
Node id
n-209b4fb94fa1da6d · revision 0 · JSON
Definition source
derived:shift_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:               1764,                1848,                  484;
  1:              74088,              116424,                60984,                10648;
  2:            3111696,             6519744,              5122656,              1788864,               234256;
  3:          130691232,           342286560,            358585920,            187830720,             49193760,              5153632;
  4:         5489031744,         17251242624,          22590912960,          15777780480,           6198413760,           1298715264,           113379904;
  5:       230539333248,        845310888576,        1328345682048,        1159666865280,         607444548480,         190911143808,         33333691776,          2494357888;
  6:      9682651996416,      40574922651648,       74387358194688,       77929613346816,       51025342072320,       21382048106496,       5600060218368,        838104250368,        54875873536;
  7:    406671383849472,    1917165095290368,     4016917342513152,     4909565640849408,     3857515860667392,     2020603546063872,     705607587514368,     158401703319552,     20743080196608,      1207269217792;
  8:  17080198121677824,   89467704446883840,   210888160481940480,   294573938450964480,   270026110246717440,   169730697869365248,   74088796689008640,   22176238464737280,   4356046841287680,    507053071472640,    26559922791424;
  9: 717368321110468608, 4133407945446033408, 10825592238072944640, 17011644945543198720, 17821723276283351040, 13069263735941124096, 6845804814064398336, 2561355542677155840, 670831213558302720, 117129259510179840, 12270684329637888, 584318301411328;
%F Formula
The row polynomials satisfy P(n,x) = (42+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 1764 + 1848*x + 484*x^2.
%t Mathematica
P[0, x_] := 1764 + 1848*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(42 + 22*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] = 1764 + 1848*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (42 + 22*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.