A028313 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 7 + 21\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	7	21	9													37
3	14	63	81	27												185
4	28	168	351	297	81											925
5	56	420	1206	1647	1053	243										4625
6	112	1008	3672	6912	7047	3645	729									23125
7	224	2352	10368	24840	34830	28431	12393	2187								115625
8	448	5376	27792	80784	144180	161352	110079	41553	6561							578125
9	896	12096	71712	244944	530712	755244	704214	413343	137781	19683						2890625
10	1792	26880	179712	705024	1796256	3102624	3674160	2939328	1515591	452709	59049					14453125
11	3584	59136	440064	1949184	5707584	11594016	16656192	16901136	11849166	5452191	1476225	177147				72265625
12	7168	129024	1057536	5218560	17262720	40310784	68094432	83770848	74401740	46451880	19309023	4782969	531441			361328125
13	14336	279552	2502144	13609728	50181120	132409728	257121216	371824992	400116024	316108980	177973686	67493007	15411789	1594323		1806640625
14	28672	602112	5842944	34725888	141191424	415362816	911471616	1515013632	1915707024	1832566032	1304274312	668907072	233302599	49424013	4782969	9033203125

Technical details

Node id: n-eee12ab308ecad8c · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-4b06920c162cf6ed

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   2,     3;
  2:   7,    21,     9;
  3:  14,    63,    81,     27;
  4:  28,   168,   351,    297,     81;
  5:  56,   420,  1206,   1647,   1053,    243;
  6: 112,  1008,  3672,   6912,   7047,   3645,    729;
  7: 224,  2352, 10368,  24840,  34830,  28431,  12393,   2187;
  8: 448,  5376, 27792,  80784, 144180, 161352, 110079,  41553,   6561;
  9: 896, 12096, 71712, 244944, 530712, 755244, 704214, 413343, 137781, 19683;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 7 + 21*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 7 + 21*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; v[3] = 7 + 21*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (3*n - 6)*x)*v[i - 1])/(n - 2)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Elements in the 5-Pascal triangle (by row).?
  - 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

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.