Real-rooted atlas

A156992 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 9\,x + 5\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101828
21505404861176
3300016200291601749665856
475000540000145800017496007873204609920
52250000202500007290000013122000011809800042515280387233280
67875000085050000038272500009185400000124002900008928208800267846264037948861440
7315000000039690000000214326000000642978000000115736040000012499492320007499695392001928493100804250272481280
8141750000000204120000000012859560000000462944160000001041624360000001499939078400001349945170560006942575162880015620794116480535534332641280
97087500000000114817500000000826686000000000347208120000000093746192400000001687431463200000020249177558400000156207941164800007029357352416000140587147048320074974806569779200
10389812500000000701662500000000056834662500000000272806380000000000859340097000000000185617460952000000027842619142800000002863812254688000000193307327191440000077322930876576000013918127557783680011546120211745996800
11233887500000000004630972500000000004167875250000000000225065263500000000008102349486000000000020417920704720000000036752257268496000000047252902202352000000042527611982116800000025516567189270080000091859641881372288000150315777624063744001939748195573327462400
1215202687500000000003283780500000000000032509426950000000000019505656170000000000007899790748850000000000227513973566880000000004777793444904480000000073714527435669120000000829288433651277600000006634307469210220800000035825260333735192320000117246306546769720320001758694598201545804800353034171594345598156800
1310641881250000000000024902002125000000000002689416229500000000000017750147114700000000000079875662016150000000000025879714493232600000000006211131478375824000000000111800366610764832000000001509304949245325232000000015093049492453252320000000108669956345663416704000005334706947878022274560000160041208436340668236800022159551937339477140480069194697632491737238732800
14798141093750000000000020113155562500000000000023532392008125000000000001694332224585000000000000083869445116957500000000000301930002421047000000000000815211006536826900000000000167700549916147248000000000026412836611793191560000000003169540393415182987200000000285258635407366468848000000018671474317573077960960000008402163442907885082432000002326752953420645099750400002991539511540829413964800014530886502823264820133888000
Technical details
Node id
n-4547a8b17f528145 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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:            10,              18;
  2:           150,             540,             486;
  3:          3000,           16200,           29160,            17496;
  4:         75000,          540000,         1458000,          1749600,           787320;
  5:       2250000,        20250000,        72900000,        131220000,        118098000,          42515280;
  6:      78750000,       850500000,      3827250000,       9185400000,      12400290000,        8928208800,        2678462640;
  7:    3150000000,     39690000000,    214326000000,     642978000000,    1157360400000,     1249949232000,      749969539200,      192849310080;
  8:  141750000000,   2041200000000,  12859560000000,   46294416000000,  104162436000000,   149993907840000,   134994517056000,    69425751628800,   15620794116480;
  9: 7087500000000, 114817500000000, 826686000000000, 3472081200000000, 9374619240000000, 16874314632000000, 20249177558400000, 15620794116480000, 7029357352416000, 1405871470483200;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+5+(9*n+9)*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[(5*n + 5 + (9*n + 9)*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; for(n = 1, N, my(i = n + 1); v[i] = (5*n + 5 + (9*n + 9)*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.