Real-rooted atlas

A094638 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22030858
311632426048748
477232844820270438411964
55832339447492076104331843840227824
649464368936110476016737841313344471168460805027536
74662724265920162747683325526438698384248817927610112645120126097632
8484156852668864245447376636252512994251024944168832516108032137889792103219203541949920
954947520694453248383622489612108691616239261633283041128886424476445440116645022722771030016185794560110129541760
106769987209764310528625188840962333511565765600473242888978149583049647097209606753078430722857489551366118023168037158912003754836274560
1190021542401460815004161065440955456460063464083213031473369344253235928542083424897187795231942034794752198160157255687549108826112147230539776081749606400139286411703040
12128524043520231966687052819007388604224933521275143043054348694632966993910432854401144725738445120134154355590835211070029215774726173670694133762140720154542083836006498304019619905536005584666976116480
13196123915008038996683909632354980587114368195633065413952072712301253936641920744520574387236995244652273024523984721267119365425644076935116840192304771397632203769974177218566488133948407808107584131170304051011754393600240665391247411200
14318595794048006923386588953606936332678895744424243294901256961767550464660805125298431430455313921176015709597003392195792797162311872024473879785993441282272375601338429440152854687940231168071088234993677107220936064835937894432317062166609920142832912302080011092925580064921600
Technical details
Node id
n-dde34e544b7b1f58 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a79b4086339383d3

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:        4,         2;
  2:       20,        30,          8;
  3:      116,       324,        260,          48;
  4:      772,      3284,       4820,        2704,         384;
  5:     5832,     33944,      74920,       76104,       33184,        3840;
  6:    49464,    368936,    1104760,     1673784,     1313344,      471168,       46080;
  7:   466272,   4265920,   16274768,    33255264,    38698384,    24881792,     7610112,      645120;
  8:  4841568,  52668864,  245447376,   636252512,   994251024,   944168832,   516108032,   137889792,   10321920;
  9: 54947520, 694453248, 3836224896, 12108691616, 23926163328, 30411288864, 24476445440, 11664502272, 2771030016, 185794560;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (2*n^2 - 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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (2*n^2 - 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.