Real-rooted atlas

A038284 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(26 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 676 + 1248\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
067612485762500
117576486724492813824125000
24569761687296233625614376963317766250000
3118813765483712010123776093450240431308807962624312500000
43089157761710918144394827264048594124803364208640124216934419110297615625000000
5803181017651897850368143717124096221103267840204095324160113037410304347807416324586471424781250000000
6208827064576154210755379249821936353289197895942144106129568563207837260447744361719712972895398605619211007531417639062500000000
75429503678976451066459484161665476158095363587179417436164966863808757764584797361930242821413761187841116163685744642575762351718426418075402241953125000000000

All 15 rows

Technical details
Node id
n-5327121044655cf2 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-42b2d489dcb16cdd

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:              676,              1248,                576;
  1:            17576,             48672,              44928,              13824;
  2:           456976,           1687296,            2336256,            1437696,             331776;
  3:         11881376,          54837120,          101237760,           93450240,           43130880,             7962624;
  4:        308915776,        1710918144,         3948272640,         4859412480,         3364208640,          1242169344,           191102976;
  5:       8031810176,       51897850368,       143717124096,       221103267840,       204095324160,        113037410304,         34780741632,         4586471424;
  6:     208827064576,     1542107553792,      4982193635328,      9197895942144,     10612956856320,       7837260447744,       3617197129728,       953986056192,       110075314176;
  7:    5429503678976,    45106645948416,    166547615809536,    358717941743616,    496686380875776,     458479736193024,     282141376118784,    111616368574464,     25757623517184,     2641807540224;
  8:  141167095653376,  1303080882954240,   5412797513809920,  13323809264762880,  21523076504616960,   23840946282037248,   18339189447720960,   9673418609786880,   3348491057233920,   686869960458240,    63403380965376;
  9: 3670344486987776, 37268113252491264, 172006676549959680, 476326181215272960, 879371411474350080, 1136418439443775488, 1049001636409638912, 691649430599761920, 319222814122967040, 98222404345528320, 18133366956097536, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (26+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 1248*x + 576*x^2.
%t Mathematica
P[0, x_] := 676 + 1248*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 676 + 1248*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 24*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 - 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.