A096956 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 6\)

\(P_{1}(x) = 7 + 18\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	6															6
1	7	18														25
2	14	57	54													125
3	28	156	279	162												625
4	56	396	1026	1161	486											3125
5	112	960	3240	5400	4455	1458										15625
6	224	2256	9360	20520	25110	16281	4374									78125
7	448	5184	25488	69120	111780	107892	57591	13122								390625
8	896	11712	66528	214704	430920	551124	438858	199017	39366							1953125
9	1792	26112	168192	628992	1505952	2395008	2531088	1714608	675783	118098						9765625
10	3584	57600	414720	1762560	4898880	9307872	12247200	11022480	6495390	2263545	354294					48828125
11	7168	125952	1002240	4769280	15085440	33312384	52418016	58786560	46058220	24013260	7499223	1062882				244140625
12	14336	273408	2382336	12545280	44478720	111881088	204773184	274827168	268476120	186201180	87038226	24623433	3188646			1220703125
13	28672	589824	5584896	32237568	126593280	357198336	745189632	1163973888	1361433744	1177830720	732679992	310361544	80247591	9565938		6103515625
14	57344	1265664	12939264	81229824	349899264	1094176512	2561974272	4563516672	6214789152	6439962672	4998852144	2818763064	1091579814	259874649	28697814	30517578125

Technical details

Node id: n-33409953afd54e79 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-8f27a5d796094068

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:    6;
  1:    7,    18;
  2:   14,    57,     54;
  3:   28,   156,    279,    162;
  4:   56,   396,   1026,   1161,     486;
  5:  112,   960,   3240,   5400,    4455,    1458;
  6:  224,  2256,   9360,  20520,   25110,   16281,    4374;
  7:  448,  5184,  25488,  69120,  111780,  107892,   57591,   13122;
  8:  896, 11712,  66528, 214704,  430920,  551124,  438858,  199017,  39366;
  9: 1792, 26112, 168192, 628992, 1505952, 2395008, 2531088, 1714608, 675783, 118098;

%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) = 6, P(1,x) = 7 + 18*x.

%t Mathematica

P[0, x_] := 6;
P[1, x_] := 7 + 18*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] = 6; v[2] = 7 + 18*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 Pascal (1,6) triangle.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.