A029635 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2 - 3\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	6														9
2	6	21	18													45
3	12	60	99	54												225
4	24	156	378	405	162											1125
5	48	384	1224	1944	1539	486										5625
6	96	912	3600	7560	8910	5589	1458									28125
7	192	2112	9936	25920	40500	37908	19683	4374								140625
8	384	4800	26208	81648	158760	197316	153090	67797	13122							703125
9	768	10752	66816	241920	562464	870912	898128	594864	229635	39366						3515625
10	1536	23808	165888	684288	1850688	3429216	4408992	3884112	2243862	767637	118098					17578125
11	3072	52224	403200	1866240	5754240	12410496	19105632	20995200	16140060	8266860	2539107	354294				87890625
12	6144	113664	963072	4942080	17107200	42083712	75442752	99307296	95265720	64953900	29878794	8325909	1062882			439453125
13	12288	245760	2267136	12773376	49040640	135489024	277136640	424942848	488453328	415704960	254619288	106288200	27103491	3188646		2197265625
14	24576	528384	5271552	32348160	136401408	418099968	960740352	1681295616	2251735200	2296769904	1756353456	976434264	373071582	87687765	9565938	10986328125

Technical details

Node id: n-4446bd9115307164 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2960fac3b919c3a3

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:   2;
  1:   3,     6;
  2:   6,    21,    18;
  3:  12,    60,    99,     54;
  4:  24,   156,   378,    405,    162;
  5:  48,   384,  1224,   1944,   1539,    486;
  6:  96,   912,  3600,   7560,   8910,   5589,   1458;
  7: 192,  2112,  9936,  25920,  40500,  37908,  19683,   4374;
  8: 384,  4800, 26208,  81648, 158760, 197316, 153090,  67797,  13122;
  9: 768, 10752, 66816, 241920, 562464, 870912, 898128, 594864, 229635, 39366;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (3*n - 3)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (3*n - 3)*x)*v[i - 1])/(n - 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from The (1,2)-Pascal triangle (or Lucas triangle) read by rows.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A038220 → Pₙ + Pₙ₊₁

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.