Real-rooted atlas

A143491 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 2\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 20 + 18\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02018442
1120148608336
28401276716176163024
3672011888828028404803230240
460480120432982964212010000124864332640
5604800132528012238246177921842403248031361283991680
6665280015787680161126249243360326222472576099456768025651891840
7798336002027577602249268481431455685763340815233568264499229107218432512726485760

All 15 rows

Technical details
Node id
n-b70ac42dd252f00b · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e7fd69580c549563

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:          20,          18,           4;
  1:         120,         148,          60,           8;
  2:         840,        1276,         716,         176,          16;
  3:        6720,       11888,        8280,        2840,         480,         32;
  4:       60480,      120432,       98296,       42120,       10000,       1248,         64;
  5:      604800,     1325280,     1223824,      617792,      184240,      32480,       3136,       128;
  6:     6652800,    15787680,    16112624,     9243360,     3262224,     725760,      99456,      7680,      256;
  7:    79833600,   202757760,   224926848,   143145568,    57633408,   15233568,    2644992,    291072,    18432,     512;
  8:  1037836800,  2795518080,  3329564544,  2310746080,  1035525440,  313303200,   64852032,   9073920,   821760,   43520,   1024;
  9: 14529715200, 41212926720, 52204939776, 39009574208, 19118848320, 6457295680, 1534534848, 256738944, 29652480, 2252800, 101376, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (n+5+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 20 + 18*x + 4*x^2.
%t Mathematica
P[0, x_] := 20 + 18*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 5 + 2*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] = 20 + 18*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 5 + 2*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.