Real-rooted atlas

A145324 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 8\,n x - 3\,n^{2} - 16\,n^{2} x - 16\,n^{2} x^{2} - n^{3} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,n + 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 11 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111819
28026096436
359236686176153611972
447524448814177616345630720385192
54190453087225412965520640487526473728014247256
640579265230404207561613674656022686822416304947220643840596312544
7429724884032928685616352299321248074076579849981194240606398054466060288027880594656
84950432011431992961131985708862362206976206834168064414822379520471794171904248691032064237817036801440798222912
961712352016462961856192156587328128240659225653573791895041437354322944024291925213184239448643338241116050817024095126814720081571131548352
108283651840250984242432337929689779226545716387584134145425988608452665942828032102193519838003214948757264793601301860301930496544333375733760418557984768005021856050996736
1111918351232040465296898566180866550144055942579376614433236022136985601355033033585971238430001605369856750057040738385929683254439156121675602847143559168286812057423052802009078326886400334060714005548544
121830471713280688845983984641177932812779008120832979368637448263378603206348839569476008393728013545462594724044803317932943933931520570980268437379481666027812636275507204675114827478204416162402186817437696010447207299809280023880332411994110976
1329898067852800123584081571072023405265745261056268575540392701440207999718760538419211455646680081713152460214626740654530561359536173056315883522933855616564236124164518811885898364354564735489963205811240963069067499244418498569836274180062969856058504360878931968001825739644772211581952
14517599316684800233219909626675204848325984180776966157563940053688320533091106938750545923322672197351697080321534209163457843961856531522575170203734835213823703140336909418496266811012261438197596163721615553712640072089635676760459719387119616213266332237576745779206346397650717664870400351026165273591808000148663454884793125466112
Technical details
Node id
n-fc1e3b5fdb7a13e5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a255b94f80393b9c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:        11,           8;
  2:        80,         260,           96;
  3:       592,        3668,         6176,          1536;
  4:      4752,       44488,       141776,        163456,         30720;
  5:     41904,      530872,      2541296,       5520640,       4875264,         737280;
  6:    405792,     6523040,     42075616,     136746560,     226868224,      163049472,       20643840;
  7:   4297248,    84032928,    685616352,    2993212480,    7407657984,     9981194240,     6063980544,      660602880;
  8:  49504320,  1143199296,  11319857088,   62362206976,  206834168064,   414822379520,   471794171904,   248691032064,    23781703680;
  9: 617123520, 16462961856, 192156587328, 1282406592256, 5357379189504, 14373543229440, 24291925213184, 23944864333824, 11160508170240, 951268147200;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(8*n^3+16*n^2+8*n)*x+(16*n^3+16*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(8*n^2+8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 11 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 11 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 11 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (8*n^3 + 16*n^2 + 8*n)*x + (16*n^3 + 16*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (8*n^2 + 8*n)*x)*v[i - 1])/(n)); 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

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.