Real-rooted atlas

A094638 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(x + 4\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
23228666
3384464184241056
4614493445264130412022176
51228802237441613445766410224720576576
6294912062300165438464251334464902488848504017873856
78257536019803340820211712011388134438279424767993685190440320643458816
82642411520708024729682500485125463257088224987366459027276896380352895737636288026381811456
995126814720281313017856367804219392279177740288135628022784437485562889372420352128626905610263744036288001213563326976
103805072588800122989156761601780661197209615212956024832849607605452832418505021448561309332481545473861121825445721612741638403991680061891729675776
1116742319390720058681316081664093107791488614488304940875776055638281869721624459433474867276571967088640170736561879042657764750848275116695552170463052804790016003465936861843456
1280363133075456003034353323999232052320311005151232544903845138923523818601761131724818973504711000064685518077198745618149710691717123495302384834564775654314752043947396956162445940454406227020800211422148572450816
13417888291992371200169037211478523904031454656376277565443565984348794519552274853829898298982415212264915304448006220984660973486081903510264047534084358516736954368073767635824394248971180682373127424524610150437481217177608717829120013953861805781753856
1423401744351572787200100929162807858954240201501657428932952064246877108096909443072207407909974965223424126416758010449756160576559114744081940481999113447012641996852960347687657472001066876271159762944160890065557880832176145048052439041323573507717120611038100736001307674368000990724188210504523776
Technical details
Node id
n-bb62cc8228f1a151 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c54a9b6c897ea5d0

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:           1;
  1:           4,            2;
  2:          32,           28,            6;
  3:         384,          464,          184,           24;
  4:        6144,         9344,         5264,         1304,          120;
  5:      122880,       223744,       161344,        57664,        10224,         720;
  6:     2949120,      6230016,      5438464,      2513344,       649024,       88848,       5040;
  7:    82575360,    198033408,    202117120,    113881344,     38279424,     7679936,     851904,      40320;
  8:  2642411520,   7080247296,   8250048512,   5463257088,   2249873664,   590272768,   96380352,    8957376,    362880;
  9: 95126814720, 281313017856, 367804219392, 279177740288, 135628022784, 43748556288, 9372420352, 1286269056, 102637440, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+(n+1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n + (n + 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n + (n + 1)*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.