Real-rooted atlas

A143491 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 4\,x + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 7\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0171220
11124760120
2118119342360840
31252451175275425206720
4133445313512154245522016060480
5142742714040369133938241128181440604800
61521162145601117695376281580508259272018144006652800
71631734273422719291767087749441619978308303343201995840079833600

All 15 rows

Technical details
Node id
n-6f34481dddb13461 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a7859c04dfd745fc

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: 1,  7,   12;
  1: 1, 12,   47,    60;
  2: 1, 18,  119,   342,     360;
  3: 1, 25,  245,  1175,    2754,     2520;
  4: 1, 33,  445,  3135,   12154,    24552,    20160;
  5: 1, 42,  742,  7140,   40369,   133938,   241128,    181440;
  6: 1, 52, 1162, 14560,  111769,   537628,  1580508,   2592720,    1814400;
  7: 1, 63, 1734, 27342,  271929,  1767087,  7494416,  19978308,   30334320,   19958400;
  8: 1, 75, 2490, 48150,  600033,  5030235, 28699460, 109911300,  270074016,  383970240,  239500800;
  9: 1, 88, 3465, 80520, 1225983, 12830664, 94092515, 483004280, 1698920916, 3894932448, 5231113920, 3113510400;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 7*x + 12*x^2.
%t Mathematica
P[0, x_] := 1 + 7*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 4)*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] = 1 + 7*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n + 4)*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.