Real-rooted atlas

A136572 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
232481898
33848646481622058
46144184322073610368194457624
5122880460800691200518400194400291602016840
6294912013271040248832002488320013996800419904052488084707280
7825753604335206409754214401219276800914457600411505920102876480110224804150656720
826424115201585446912041617981440624269721605852528640035115171840131681894402821754880264539520232436776320
9951268147206421059993601926317998080337105649664037924385587202844328919040142216445952045712429056085710804480714256704014643516908160
1038050725888002853804441600096315899904000192631799808000252829237248000227546313523200142216445952000609499054080001714216089600028570268160002142770112001025046183571200
11167423193907200138124134973440051796550615040001165422388838400017481335832576000183554026242048001376655196815360073749385543680002765601957888000691400489472000103710073420800707114136960078928556134982400
128036313307545600723268197679104002983481315426304007458703288565760001258656179945472000151038741593456640013215889889427456008495929214631936003982466819358720001327488939786240002986850114519040040729774288896002545610893056006629998715338521600
134178882919923712004074410846925619200183348488111652864005042083423070453760094539064182571008000127627736646470860800127627736646470860800957208024848531456005384295139772989440022434563082387456000673036892471623680013766663709646848001720832963705856009927882482918400603329883095805465600
142340174435157278720024571831569151426560011978767889961320448003593630366988396134400741186263191356702720011117793947870350540800125075181913541443584001072072987830355230720070354789826367062016003517739491318353100800131915230924438241280035976881161210429440067456652177269555200778345986660802560041697106428257280059126328543388935628800
Technical details
Node id
n-307f56251024b704 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c91d5d506c107735

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,            3;
  2:          32,           48,            18;
  3:         384,          864,           648,           162;
  4:        6144,        18432,         20736,         10368,          1944;
  5:      122880,       460800,        691200,        518400,        194400,         29160;
  6:     2949120,     13271040,      24883200,      24883200,      13996800,       4199040,        524880;
  7:    82575360,    433520640,     975421440,    1219276800,     914457600,     411505920,     102876480,     11022480;
  8:  2642411520,  15854469120,   41617981440,   62426972160,   58525286400,   35115171840,   13168189440,   2821754880,   264539520;
  9: 95126814720, 642105999360, 1926317998080, 3371056496640, 3792438558720, 2844328919040, 1422164459520, 457124290560, 85710804480, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+3*n*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 + 3*n*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 + 3*n*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.