Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2549
325401681
412530024064729
56252000240012802566561
631251250020000160006400102459049
7156257500015000016000096000307204096531441
878125437500105000014000001120000537600143360163844782969
9390625250000070000001120000011200000716800028672006553606553643046721
1019531251406250045000000840000001008000008064000043008000147456002949120262144387420489
11976562578125000281250000600000000840000000806400000537600000245760000737280001310720010485763486784401
1248828125429687500171875000041250000006600000000739200000059136000003379200000135168000036044800057671680419430431381059609
132441406252343750000103125000002750000000049500000000633600000005913600000040550400000202752000007208960000173015040025165824016777216282429536481
14122070312512695312500609375000001787500000003575000000005148000000005491200000004392960000002635776000001171456000003748659200081788928001090519040671088642541865828329
15610351562568359375000355468750000113750000000025025000000004004000000000480480000000043929600000003075072000000164003840000065601536000019084083200038168166400469762048026843545622876792454961
Technical details
Node id
n-a041f87d41cb6b7b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6b726760bd05165a

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       5,        4;
   3:      25,       40,       16;
   4:     125,      300,      240,       64;
   5:     625,     2000,     2400,     1280,       256;
   6:    3125,    12500,    20000,    16000,      6400,     1024;
   7:   15625,    75000,   150000,   160000,     96000,    30720,     4096;
   8:   78125,   437500,  1050000,  1400000,   1120000,   537600,   143360,    16384;
   9:  390625,  2500000,  7000000, 11200000,  11200000,  7168000,  2867200,   655360,   65536;
  10: 1953125, 14062500, 45000000, 84000000, 100800000, 80640000, 43008000, 14745600, 2949120, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (5+4*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (5 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.