Real-rooted atlas

A038334 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 784 + 448\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0784448641296
12195218816537651246656
26146567024643010565734440961679616
317210368245862401404928040140805734403276860466176
44818903048260976645900697602247884804816896055050242621442176782336
513492928512269858570242313073459211014635520314703872053949235251380224209715278364164096
6377801998336863547424768863547424768493455671296176234168320402820956165754585088469762048167772162821109907456
71057845595340827201743880192310877072916482072513819443288822020833282537772023808483385147392591900180484227858432134217728101559956668416
829619676669542484627647627264010880697552076808290055277772804145027638886401421152333332483383696031744055244016844805919001804803758096384010737418243656158440062976
9829350946747187226065315469197312372361649559961603191671281942528018238121611100160729524864444006420843567555543044253789297049606076841852928057874684313603307124817928589934592131621703842267136
1023221826508921241679617690887729971212511351425214709761191557278591877120766001107666206720350171934933123072116723978311041024285854640761733125104547156459520648196464312320555596969410562886218022912687194767364738381338321616896
11650211142249794764824150699569278091264414011992616195850244337268494074432716830980489243388805120159328230394570997766069646872174133248173418482062118092837161103299025305658985878252421120674124322884812852529168016998425013889531904549755813888170581728179578208256
121820591198299425341447282364793197701365761352439175879573110784154564477243379784089612144351783408411607046939629590519092346882974126967365325291529711434995478613196824278587488696532992462449285498981580866064183642711654468638112875544576490272234825318421550427904409643980465111046140942214464815497216
1350976553552383909560322184709437959310409728043694188759186208194560540975670351829244313604636934317301393522688029146444280180187856896138792591810381846937605098503372626271928320145671524932179197952032371449984928710656055493914259877789696720700185193218048068638112875544576045255898599260160184717953466368035184372088832221073919720733357899776
14142734349946674946768896652499885470514042372096139821404029395866222592018642853870586115496345601731122145125853581803520118705518522915674180812862179081131051067428044825379216788184109154304081576053962020350853120207177279903543748198404143545598070874963968645747365933123371008768746864206099251206758214190822850560413768215764664320157625986957967362814749767106567958661109946400884391936
Technical details
Node id
n-01f9460eada72015 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8d10fbbacee008e8

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:              784,               448,                64;
  1:            21952,             18816,              5376,               512;
  2:           614656,            702464,            301056,             57344,              4096;
  3:         17210368,          24586240,          14049280,           4014080,            573440,            32768;
  4:        481890304,         826097664,         590069760,         224788480,          48168960,          5505024,           262144;
  5:      13492928512,       26985857024,       23130734592,       11014635520,        3147038720,        539492352,         51380224,         2097152;
  6:     377801998336,      863547424768,      863547424768,      493455671296,      176234168320,      40282095616,       5754585088,       469762048,       16777216;
  7:   10578455953408,    27201743880192,    31087707291648,    20725138194432,     8882202083328,    2537772023808,     483385147392,     59190018048,     4227858432,     134217728;
  8:  296196766695424,   846276476272640,  1088069755207680,   829005527777280,   414502763888640,  142115233333248,   33836960317440,   5524401684480,   591900180480,   37580963840,   1073741824;
  9: 8293509467471872, 26065315469197312, 37236164955996160, 31916712819425280, 18238121611100160, 7295248644440064, 2084356755554304, 425378929704960, 60768418529280, 5787468431360, 330712481792, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = (28+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 448*x + 64*x^2.
%t Mathematica
P[0, x_] := 784 + 448*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 8*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] = 784 + 448*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 8*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.