Real-rooted atlas

A088996 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 12\,n x - 4\,n x^{2} + 12\,n^{2} + 20\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
23202447
315136300192643
4105119237324544192011493
59451282851468906567408023040253017
6103951637887970521820704217472013201923225606609411
7135135242030413801224385474566013368053697792257564165160960199652967
8202702540628880265105656872637952167295432019600335361388705024548388864928972806843378537
93445942576366386056037759122119738092848074577360694563834246476868966437890256896126841282561857945600262331261325
1065472907515887990340129381658128552184261952144335149304024734611416962851254050176220034207334410935292170243171704832004087480320011118091901175
1113749310575362455876440324131850190815391064806080454954485575209004716339187212336183769056011789585680179277422579872768333912943411208533896560640980995276800516137660988075
1231623414322589964520657208759873709572445771668260089615081272610053603381353676469632535864284185529660952436830965764963907470722048283260189978624010778097915002882459958194995202550587719680026043816427037325
137905853580625241355477090700254083005795618014479673284340832525789737096631201315874124551080322365337992326279043116093365211146243023835644986260482142168188710707201079576591213936643672050048276889675665291791564807141645615104001419138523306008225
1421345804667687569593256819639007873934160262626048580883417423318419264208551128333605319513652932597696106849347194811491201598253366151418777617980050341878013952151913427828094976009512734817506607104428906928256096665613181514691478814722474423999476531202142493684531200083045339879242200075
Technical details
Node id
n-fb39c381046991b5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7e3e6d2fe95370dc

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:        1,         4;
  2:        3,        20,         24;
  3:       15,       136,        300,         192;
  4:      105,      1192,       3732,        4544,        1920;
  5:      945,     12828,      51468,       90656,       74080,       23040;
  6:    10395,    163788,     797052,     1820704,     2174720,     1320192,      322560;
  7:   135135,   2420304,   13801224,    38547456,    60133680,    53697792,    25756416,     5160960;
  8:  2027025,  40628880,  265105656,   872637952,  1672954320,  1960033536,  1388705024,   548388864,    92897280;
  9: 34459425, 763663860, 5603775912, 21197380928, 48074577360, 69456383424, 64768689664, 37890256896, 12684128256, 1857945600;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(8*n^3-20*n^2+12*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(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.