Real-rooted atlas

A008955 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 4\,n^{2} x - n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x + 4\,n^{3} x^{2} - 8\,n^{4} - 14\,n^{4} x - 6\,n^{4} x^{2} + 4\,n^{5} + 8\,n^{5} x + 4\,n^{5} x^{2} - n^{6} - 2\,n^{6} x - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 2\,n^{2} x - 4\,n^{3} - 4\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
210261248
31004404711441155
41700106801856712496288046323
544200362680883242943796465380864002785698
6163540016601560522646747731606859689880233012163628800234437598
7817700009903472003833448360728042876877201031654674825744151349284820321280026297628885
85315050000748391280003442474746008003487966401084241007165896767596432447430690672123910023168146313216003791731088277
943583410000069978465960003732124696920010280732858128016994307824733017892316880386212143385449792851579920121984124910692323841316818944000683250166093968
10440192441000007939493261960004819122933289200153597064492692803001516697564033038464409438869662331391889980746281908772673208358470643390061939841520895294720000144850083840000150453374482276668
11537034778020000010751447486811200073203480056585640026513693533815097605985005971480098860905086019640507468094728735235522052356911379082762231520346016616016054392011353103590108089602201028739557120001912021106688000039751107354989233335
12778700428129000000171362590165738400001293681726535613700005249992379489633440001345072349473286091500234658376035049691160828941111739111223053152561114295447972203360162136750463076037464071824658768225432640021174807759089037926437348591167922176000298275292643328000012410857463228365602087
131323790727819300000003176364777525154800000263366310118069013400001184008320225929305200003395683884271771419750006716999117327668943295609546669976473304329755949943663312444177541120407625643312092749788526204265587068866482391704001695192138405824437526404538790607621274166067273449725638207426560005428610326108569600004520265827545549143104526
142607867733804021000000067763645770297205560000061221675539183550751800003020760347953634419596000095903176365689164946475000212191367569427141975443320341664778352750125132884738409735574728152314546604936369482949287621285615670960250101630720022920733361600125361342331356101784408320451877075836548108077168641108432498525519178251468816572978322800401940480001140008168482799616000001899517636741553089017468426
Technical details
Node id
n-d49f9ac2a064675e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d6fc3f107f959e08

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:            2,             2;
  2:           10,            26,             12;
  3:          100,           440,            471,             144;
  4:         1700,         10680,          18567,           12496,            2880;
  5:        44200,        362680,         883242,          943796,          465380,           86400;
  6:      1635400,      16601560,       52264674,        77316068,        59689880,        23301216,         3628800;
  7:     81770000,     990347200,     3833448360,      7280428768,      7720103165,      4674825744,      1513492848,      203212800;
  8:   5315050000,   74839128000,   344247474600,    800348796640,   1084241007165,    896767596432,    447430690672,   123910023168,    14631321600;
  9: 435834100000, 6997846596000, 37321246969200, 102807328581280, 169943078247330, 178923168803862, 121433854497928, 51579920121984, 12491069232384, 1316818944000;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(2*n^6-8*n^5+14*n^4-12*n^3+4*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*n + 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

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.