Real-rooted atlas

A145324 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
212341864
36026629496716
43602196375925366009451
52520196924607747824232204320143653
620160192816575382812644604220229608352802470110
71814402057904749223013451204135644207866984246052832256047397270
81814400238449601027040042244188002869702452251655761072086122853388832659201003926405
9199584002985825601487418444383799352059619076955920464276380622645215391140483569313603628800023264884755
102395008004022075520227756335686787139275212431530324014974275288612165400896666319072544233297126644797950400439084800585506488140
1131135104005803500096036848059574412467382428642634169833400373679615417436779006569622530545780320119859374397637343619072069041047680574801920015902598776400
124358914560089344128384062904108591362384133239299257137874114140934081955594761081653508691079007073639934453789544452844225411274164006306712194816105959985408080951270400463628865812175
136538371840001462211533440011311842984768047504098859875212742914285894602361039218509692314814745717471330817066598958722225457265082328117484315829024044173404513480011221469315020817284866842880122049607680014441374659711825
1410461394944000253568960870400213889247258688098630244145336322929568139365796060713873210765352916885456387150181032755486813290328761111612706008855888444950682640264526568339457609023810374427328210004663317696029874501699840019615115520000478634031219213450
Technical details
Node id
n-b6060bc044d672a9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-360a628193aab25f

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:        3,         4;
  2:       12,        34,         18;
  3:       60,       266,        294,         96;
  4:      360,      2196,       3759,       2536,        600;
  5:     2520,     19692,      46077,      47824,      23220,       4320;
  6:    20160,    192816,     575382,     812644,     604220,     229608,      35280;
  7:   181440,   2057904,    7492230,   13451204,   13564420,    7866984,    2460528,     322560;
  8:  1814400,  23844960,  102704004,  224418800,  286970245,  225165576,  107208612,   28533888,   3265920;
  9: 19958400, 298582560, 1487418444, 3837993520, 5961907695, 5920464276, 3806226452, 1539114048, 356931360, 36288000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(2*n^3+4*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (2*n^3 + 4*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (2*n^2 + 2*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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (2*n^3 + 4*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (2*n^2 + 2*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.