A097805 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	2	9	9													20
3	4	24	45	27												100
4	8	60	162	189	81											500
5	16	144	504	864	729	243										2500
6	32	336	1440	3240	4050	2673	729									12500
7	64	768	3888	10800	17820	17496	9477	2187								62500
8	128	1728	10080	33264	68040	88452	71442	32805	6561							312500
9	256	3840	25344	96768	235872	381024	408240	279936	111537	19683						1562500
10	512	8448	62208	269568	762048	1469664	1959552	1784592	1062882	373977	59049					7812500
11	1024	18432	149760	725760	2332800	5225472	8328096	9447840	7479540	3936600	1240029	177147				39062500
12	2048	39936	354816	1900800	6842880	17449344	32332608	43879968	43302600	30311820	14289858	4074381	531441			195312500
13	4096	86016	829440	4866048	19388160	55427328	117013248	184757760	218245104	190531440	119515176	51018336	13286025	1594323		976562500
14	8192	184320	1916928	12220416	53374464	169019136	400308480	720555264	990763488	1035798192	810624672	460582200	179627058	43046721	4782969	4882812500

Technical details

Node id: n-7f0d815631f41e18 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-371721bfd00b48c7

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:   1,    3;
  2:   2,    9,     9;
  3:   4,   24,    45,    27;
  4:   8,   60,   162,   189,     81;
  5:  16,  144,   504,   864,    729,    243;
  6:  32,  336,  1440,  3240,   4050,   2673,    729;
  7:  64,  768,  3888, 10800,  17820,  17496,   9477,   2187;
  8: 128, 1728, 10080, 33264,  68040,  88452,  71442,  32805,   6561;
  9: 256, 3840, 25344, 96768, 235872, 381024, 408240, 279936, 111537, 19683;

%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) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*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] = 1; v[2] = 1 + 3*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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read b…?
  - 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

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.