Real-rooted atlas

A178820 → basis A013611 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(18 + 48\,x + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246488
2360192025604840
34320345609216081920212960
4453604838401935360344064022937608198960
54354565806080309657608257536011010048058720256288603392
639191046270566441803776014863564802972712960317089382414092861449523911936
73359232062705664050164531202229534720059454259200951268147208455716864032212254720299322946560

All 15 rows

Technical details
Node id
n-5769a70eea27be08 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-10cba46fcb583bff

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 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:         24,          64;
  2:        360,        1920,         2560;
  3:       4320,       34560,        92160,         81920;
  4:      45360,      483840,      1935360,       3440640,        2293760;
  5:     435456,     5806080,     30965760,      82575360,      110100480,       58720256;
  6:    3919104,    62705664,    418037760,    1486356480,     2972712960,     3170893824,     1409286144;
  7:   33592320,   627056640,   5016453120,   22295347200,    59454259200,    95126814720,    84557168640,    32212254720;
  8:  277136640,  5912248320,  55180984320,  294298583040,   980995276800,  2092789923840,  2790386565120,  2126008811520,   708669603840;
  9: 2217093120, 53210234880, 567575838720, 3531582996480, 14126331985920, 37670218629120, 66969277562880, 76536317214720, 51024211476480, 15118284881920;
%F Formula
The row polynomials satisfy n*P(n,x) = (6*n+18+(16*n+48)*x)*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[((6*n + 18 + (16*n + 48)*x)*P[n - 1, x])/(n)]];
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] = ((6*n + 18 + (16*n + 48)*x)*v[i - 1])/(n)); 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.