Real-rooted atlas

A139548 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 3\,x + 6\,n\right)\,P_{n}(x) = \left(16 + 72\,x + 108\,x^{2} + 54\,x^{3} + 48\,n + 144\,n x + 108\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1326018110
24482112345621603248500
3512039168120960190080155520583205832575000
453248565248258048065802241016064095800325225472139968010497636250000
55242887127040431308801526169603483648005329981445486745603695155201511654403149280018895682187500000
6498073682575360622854144282230784085399142401813089484827590492160301524664322327947776012194012160399076761668024448034012224128125000000
74613734490282393681584455684508614656017003918131246229402419293255863500814142097981441614043791360137193722265684737299046436315985305699043596288142851340806122200327343750000000

All 15 rows

Technical details
Node id
n-c5c61f28a5dc4bc9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7939c1a278d41502

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:          1;
  1:         32,          60,            18;
  2:        448,        2112,          3456,          2160,            324;
  3:       5120,       39168,        120960,        190080,         155520,           58320,            5832;
  4:      53248,      565248,       2580480,       6580224,       10160640,         9580032,         5225472,          1399680,           104976;
  5:     524288,     7127040,      43130880,     152616960,      348364800,       532998144,       548674560,        369515520,        151165440,         31492800,          1889568;
  6:    4980736,    82575360,     622854144,    2822307840,     8539914240,     18130894848,     27590492160,      30152466432,      23279477760,      12194012160,       3990767616,        680244480,         34012224;
  7:   46137344,   902823936,    8158445568,   45086146560,   170039181312,    462294024192,    932558635008,    1414209798144,    1614043791360,    1371937222656,     847372990464,     363159853056,      99043596288,      14285134080,       612220032;
  8:  419430400,  9462349824,   99656663040,  650033233920,  2937040404480,   9738607656960,  24485642108928,   47550242488320,   71940237557760,   84852587888640,   77474101985280,   53879716380672,   27732206960640,   10056734392320,   2350924922880,    293865615360,    11019960576;
  9: 3758096384, 96032784384, 1155111321600, 8686437138432, 45742408335360, 179019152621568, 539240618262528, 1277148832727040, 2408337798856704, 3637593550356480, 4402742538534912, 4248800491732992, 3232782982840320, 1902368447594496, 839280197468160, 263773776347136, 53953726980096, 5950778711040, 198359290368;
%F Formula
The row polynomials satisfy (6*n-4+3*x)*P(n,x) = (48*n+16+(144*n+72)*x+(108*n+108)*x^2+54*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((48*n + 16 + (144*n + 72)*x + (108*n + 108)*x^2 + 54*x^3)*P[n - 1, x])/(6*n - 4 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((48*n + 16 + (144*n + 72)*x + (108*n + 108)*x^2 + 54*x^3)*v[i - 1])/(6*n - 4 + 3*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.