Real-rooted atlas

A156992 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 12\,x + 6\,x^{2} + x^{3} + 8\,n + 24\,n x + 16\,n x^{2} + 3\,n x^{3} + 12\,n^{2} x + 12\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2249654174
3192115212963843024
41920153602592015360300061560
523040230400518400460800180000259201438560
6322560387072010886400129024007560000217728024696037966320
75160960722534402438553603612672002822400001219276802765952025804801116944640
892897280148635648058525286401040449536010160640000585252864019914854403715891202939328036241914240
91857945600334430208001504935936003121348608003657830400002633637888001194891264003344302080052907904003628800001285662067200
10408748032008174960640004138573824000980995276800013412044800000115880067072006571901952000245248819200058198694400079833600000482993280049497989587200
119809952768002158189608960012139816550400032372844134400050582568960000050987229511680034699642306560016186422067200051214851072000105380352000001275102259200689762304002055345091430400
1225505877196800612141052723200378762276372480011222585966592000197272018944000002272573658234880018043813999411200101003273699328003994758383616000109595566080000019891595243520021520583884800105236651520091557138453580800
1371416456151040018568278599270400125335880545075200408502129183948800797855721062400000103402101449687040093827832796938240061275319377592320029081841032724480099731965132800000241351355621376003916746267033600383061411532800170869450752004355031115840204800
1421424936845312000599898231668736000438675581907763200015597354023387136000335099402846208000004825431400985395200049259612218392576000367651916265553920002035728872290713600083774850711552000002534189234024448000548344477384704000804428964218880007176516931584000294226732800000220299712295959296000
Technical details
Node id
n-8b60a69ef260fa3b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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,           8;
  2:         24,          96,           54;
  3:        192,        1152,         1296,          384;
  4:       1920,       15360,        25920,        15360,         3000;
  5:      23040,      230400,       518400,       460800,       180000,        25920;
  6:     322560,     3870720,     10886400,     12902400,      7560000,      2177280,       246960;
  7:    5160960,    72253440,    243855360,    361267200,    282240000,    121927680,     27659520,     2580480;
  8:   92897280,  1486356480,   5852528640,  10404495360,  10160640000,   5852528640,   1991485440,   371589120,   29393280;
  9: 1857945600, 33443020800, 150493593600, 312134860800, 365783040000, 263363788800, 119489126400, 33443020800, 5290790400, 362880000;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (8*n+8+(12*n^2+24*n+12)*x+(2*n^3+12*n^2+16*n+6)*x^2+(n^3+3*n^2+3*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 + (12*n^2 + 24*n + 12)*x + (2*n^3 + 12*n^2 + 16*n + 6)*x^2 + (n^3 + 3*n^2 + 3*n + 1)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 8 + (12*n^2 + 24*n + 12)*x + (2*n^3 + 12*n^2 + 16*n + 6)*x^2 + (n^3 + 3*n^2 + 3*n + 1)*x^3)*v[i - 1])/(4 + (6*n - 2)*x + n^2*x^2)); 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

Same fingerprint ?

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.