Real-rooted atlas

A038281 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(112 + 66\,x + 9\,x^{2} + 24\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 112 + 66\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0112669187
110881008297272420
21024013056604811888130613
39420815360097920302404455243380666
485196816957441382400587520136080160387294670479
5760217617891328178053129676800308448057153656133218756689952
667108864182452224214695936142442496580608001480550422861441924566561682050985
7587202560181193932824631050241932263424961486848313528320666247688817984649539196838145637478

All 15 rows

Technical details
Node id
n-9f5c546bbf0f2937 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bd3dad0f6f053f13

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:         112,           66,            9;
  1:        1088,         1008,          297,           27;
  2:       10240,        13056,         6048,         1188,           81;
  3:       94208,       153600,        97920,        30240,         4455,         243;
  4:      851968,      1695744,      1382400,       587520,       136080,       16038,         729;
  5:     7602176,     17891328,     17805312,      9676800,      3084480,      571536,       56133,       2187;
  6:    67108864,    182452224,    214695936,    142442496,     58060800,    14805504,     2286144,     192456,       6561;
  7:   587202560,   1811939328,   2463105024,   1932263424,    961486848,   313528320,    66624768,    8817984,     649539,     19683;
  8:  5100273664,  17616076800,  27179089920,  24631050240,  14491975680,  5768921088,  1567641600,  285534720,   33067440,   2165130,   59049;
  9: 44023414784, 168309030912, 290665267200, 298969989120, 203206164480, 95647039488, 31729065984, 7390310400, 1177830720, 121247280, 7144929, 177147;
%F Formula
The row polynomials satisfy (3*n+11+3*x)*P(n,x) = (24*n+112+(9*n+66)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 112 + 66*x + 9*x^2.
%t Mathematica
P[0, x_] := 112 + 66*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 112 + (9*n + 66)*x + 9*x^2)*P[n - 1, x])/(3*n + 11 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 112 + 66*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 112 + (9*n + 66)*x + 9*x^2)*v[i - 1])/(3*n + 11 + 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.