Real-rooted atlas

A272866 → D → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 10\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1549
25014012032342
337518003300288012001929747
42500170004800072800640003264089601024246924
5156251375005250001140000155000013680007840002816005760051205864445
6937501012500487500013800000255000003228000028560000176640007488000207360033792024576133709346
7546875700000040687500142100000332500000550200000662200000586880000383040000181888000611072001376256018636801146882963890503

All 15 rows

Technical details
Node id
n-dff119f470326354 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-dc7db2765c5515ed

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        5,         4;
  2:       50,       140,        120,          32;
  3:      375,      1800,       3300,        2880,        1200,         192;
  4:     2500,     17000,      48000,       72800,       64000,       32640,         8960,         1024;
  5:    15625,    137500,     525000,     1140000,     1550000,     1368000,       784000,       281600,        57600,         5120;
  6:    93750,   1012500,    4875000,    13800000,    25500000,    32280000,     28560000,     17664000,      7488000,      2073600,       337920,       24576;
  7:   546875,   7000000,   40687500,   142100000,   332500000,   550200000,    662200000,    586880000,    383040000,    181888000,     61107200,    13762560,     1863680,      114688;
  8:  3125000,  46250000,  315000000,  1309000000,  3710000000,  7593600000,  11592000000,  13436800000,  11923200000,   8099840000,   4178944000,  1608499200,   447283200,    84869120,    9830400,    524288;
  9: 17578125, 295312500, 2306250000, 11115000000, 37012500000, 90342000000, 167328000000, 240163200000, 270396000000, 240163200000, 168284160000, 92510208000, 39416832000, 12747571200, 3022848000, 495452160, 50135040, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+10*n*x+4*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 10*n*x + 4*n*x^2)*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] = 0; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 10*n*x + 4*n*x^2)*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

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.