Real-rooted atlas

A034870 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
1246
2424361680
366018024015036672
48112504112014001008392644608
51018010803360630075605880288081010028160
61226419807920198003326438808316801782066001452144159744
7143643276160165005010810816816819219216216210010044044131042366196860160
816480504029120109200288288560560823680926640800800528528262080946402352036002564456448
918612734448960214200668304155937628005123938220437580038507042673216144799259976018360039168520232422413312
102076010260775203876001395360379848080620801360476018475600203231601813968013100880759696034884001240320329460615607220400110100480
1122924138601170406583502686068835665620465280402910206466460085357272931170248406398062674920383724001910092876105262370060555940924009702484530579456
122411041821617001610626004845456169590964707014410590782419612560030203342438939846442184833638440617629418840018828057610002405643609104153439444250400892584133584126965762516582400
1326130023400239200164450082882803223220099985600253088550531173500934865360139070880017577014001892909200173838600013598041609029949505061771002374658009209200029008980723580013754001872001625067611777605632
14281512294843276002457000135626405801796019891872055945890013123110002598375780438073272063277250407862853600842448600077879692806206038020425188764024933909001243242000522161640182342160519901201179360020475002555282041278454492397568
Technical details
Node id
n-477fe85c0991ab04 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6fde533e0acf53df

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  2,   4;
  2:  4,  24,   36,    16;
  3:  6,  60,  180,   240,    150,     36;
  4:  8, 112,  504,  1120,   1400,   1008,     392,      64;
  5: 10, 180, 1080,  3360,   6300,   7560,    5880,    2880,     810,     100;
  6: 12, 264, 1980,  7920,  19800,  33264,   38808,   31680,   17820,    6600,    1452,     144;
  7: 14, 364, 3276, 16016,  50050, 108108,  168168,  192192,  162162,  100100,   44044,   13104,    2366,    196;
  8: 16, 480, 5040, 29120, 109200, 288288,  560560,  823680,  926640,  800800,  528528,  262080,   94640,  23520,   3600,   256;
  9: 18, 612, 7344, 48960, 214200, 668304, 1559376, 2800512, 3938220, 4375800, 3850704, 2673216, 1447992, 599760, 183600, 39168, 5202, 324;
%F Formula
The row polynomials satisfy (n-1+(2*n^2-4*n+2)*x)*P(n,x) = (n+(2*n^2+2*n)*x+(4*n^2+n)*x^2+2*n^2*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + (2*n^2 + 2*n)*x + (4*n^2 + n)*x^2 + 2*n^2*x^3)*P[n - 1, x])/(n - 1 + (2*n^2 - 4*n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + (2*n^2 + 2*n)*x + (4*n^2 + n)*x^2 + 2*n^2*x^3)*v[i - 1])/(n - 1 + (2*n^2 - 4*n + 2)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.