Real-rooted atlas

A038338 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(192 + 360\,x + 108\,x^{2} + 72\,n + 108\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2304 + 4320\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02304432012967920
17603224883223328046656604800
22322432109486081791590411197440167961644064000
3676823044180377609853747201074954240503884800604661763110400000
4191102976014619377664451480780807094697984058047528960217678233602176782336214617600000
55274442137648157949952018420415856643792438558720446965972992029255954595849142485811207836416409614556672000000
614309790842881519039335628869347447930880176835992223744273055576227840257452400443392140428582060032376147987660802821109907456974177280000000
738306209333248463637223309312246084372371865674895243765350401432371537012326417694001339564032139024296239431686499837226778624152339935002624010155995666841664497254400000000

All 15 rows

Technical details
Node id
n-25fbbc0c07682907 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:              2304,               4320,                1296;
  1:             76032,             248832,              233280,                46656;
  2:           2322432,           10948608,            17915904,             11197440,              1679616;
  3:          67682304,          418037760,           985374720,           1074954240,            503884800,             60466176;
  4:        1911029760,        14619377664,         45148078080,          70946979840,          58047528960,          21767823360,           2176782336;
  5:       52744421376,       481579499520,       1842041585664,        3792438558720,        4469659729920,        2925595459584,         914248581120,          78364164096;
  6:     1430979084288,     15190393356288,      69347447930880,      176835992223744,      273055576227840,      257452400443392,      140428582060032,       37614798766080,        2821109907456;
  7:    38306209333248,    463637223309312,    2460843723718656,     7489524376535040,    14323715370123264,    17694001339564032,    13902429623943168,     6499837226778624,     1523399350026240,      101559956668416;
  8:  1014454095446016,  13790235359969280,   83454700195676160,   295301246846238720,   674057193888153600,  1031307506648875008,  1061640080373841920,   714982094945648640,   292492675205038080,    60935974001049600,    3656158440062976;
  9: 26629420005457920, 401723821796622336, 2730466601273917440, 11016020425829253120, 29234823437777633280, 53385329755941765120, 68066295438825750528, 60058495975434485760, 35391613699809607680, 12869677709021675520, 2413064570441564160, 131621703842267136;
%F Formula
The row polynomials satisfy (3*n+5+3*x)*P(n,x) = (72*n+192+(108*n+360)*x+108*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2304 + 4320*x + 1296*x^2.
%t Mathematica
P[0, x_] := 2304 + 4320*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72*n + 192 + (108*n + 360)*x + 108*x^2)*P[n - 1, x])/(3*n + 5 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2304 + 4320*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72*n + 192 + (108*n + 360)*x + 108*x^2)*v[i - 1])/(3*n + 5 + 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. [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.