Real-rooted atlas

A038333 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(40 + 21\,x + 21\,n\right)\,P_{n}(x) = \left(1159 + 1680\,x + 441\,x^{2} + 399\,n + 441\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1159 + 1680\,x + 441\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0115916804413280
12960273017529209261164800
27064772486568306671414817601944817936000
31615980474180085130544820107334990388962004084101371200000
435903435520361353044673345355548288244033810521859801842408576612116998400000
578096162465277805018514965594484422899392239520149592967099402934239240145138801801088541765952000000
61671540151931312015529328443335621554083807329112649617744740590677026323691227832821586925763483331203782285936134078720000000
735325811125283159210887147712398546752149627930144157902039598963005722436355075119430221326329196272875148618284684136162293699607942800465811500774400000000

All 15 rows

Technical details
Node id
n-cdf36e798942c344 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9d4c93c35b848be1

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:             1159,              1680,               441;
  1:            29602,             73017,             52920,               9261;
  2:           706477,           2486568,           3066714,            1481760,             194481;
  3:         16159804,          74180085,         130544820,          107334990,           38896200,            4084101;
  4:        359034355,        2036135304,        4673345355,         5482882440,         3381052185,          980184240,           85766121;
  5:       7809616246,       52778050185,      149655944844,       228993922395,       201495929670,        99402934239,        24014513880,         1801088541;
  6:     167154015193,     1312015529328,     4433356215540,      8380732911264,      9617744740590,      6770263236912,      2783282158692,       576348333120,        37822859361;
  7:    3532581112528,    31592108871477,   123985467521496,    279301441579020,    395989630057224,    363550751194302,    213263291962728,     75148618284684,     13616229369960,      794280046581;
  8:   73895482791391,   741842033630880,  3317171431505085,   8678982726504720,  14663325682898550,  16631564462403408,  12724276291800570,   6397898758881840,   1972651229972955,   317712018632400,   16679880978201;
  9: 1532766564450250, 17069856524811321, 85682754884366640, 255422200225891545, 501211252455647580, 677445646549913010, 640315231802531208, 419901117629418810, 184739326662713130, 50631381569305845, 7339147630408440, 350277500542221;
%F Formula
The row polynomials satisfy (21*n+40+21*x)*P(n,x) = (399*n+1159+(441*n+1680)*x+441*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1159 + 1680*x + 441*x^2.
%t Mathematica
P[0, x_] := 1159 + 1680*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((399*n + 1159 + (441*n + 1680)*x + 441*x^2)*P[n - 1, x])/(21*n + 40 + 21*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1159 + 1680*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((399*n + 1159 + (441*n + 1680)*x + 441*x^2)*v[i - 1])/(21*n + 40 + 21*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

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.