Real-rooted atlas

A272866 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 10\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 10 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
110818
210028024064684
3750360066005760240038419494
45000340009600014560012800065280179202048493848
531250275000105000022800003100000273600015680005632001152001024011728890
618750020250009750000276000005100000064560000571200003532800014976000414720067584049152267418692
7109375014000000813750002842000006650000001100400000132440000011737600007660800003637760001222144002752512037273602293765927781006

All 15 rows

Technical details
Node id
n-cd63d144a226e1c8 · revision 0 · JSON
Definition source
derived:derivative
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:       10,         8;
  2:      100,       280,        240,          64;
  3:      750,      3600,       6600,        5760,        2400,          384;
  4:     5000,     34000,      96000,      145600,      128000,        65280,        17920,         2048;
  5:    31250,    275000,    1050000,     2280000,     3100000,      2736000,      1568000,       563200,       115200,        10240;
  6:   187500,   2025000,    9750000,    27600000,    51000000,     64560000,     57120000,     35328000,     14976000,      4147200,       675840,        49152;
  7:  1093750,  14000000,   81375000,   284200000,   665000000,   1100400000,   1324400000,   1173760000,    766080000,    363776000,    122214400,     27525120,     3727360,      229376;
  8:  6250000,  92500000,  630000000,  2618000000,  7420000000,  15187200000,  23184000000,  26873600000,  23846400000,  16199680000,   8357888000,   3216998400,   894566400,   169738240,   19660800,   1048576;
  9: 35156250, 590625000, 4612500000, 22230000000, 74025000000, 180684000000, 334656000000, 480326400000, 540792000000, 480326400000, 336568320000, 185020416000, 78833664000, 25495142400, 6045696000, 990904320, 100270080, 4718592;
%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) = 10 + 8*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 10 + 8*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] = 10 + 8*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.