A167194 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

\(P_{2}(x) = 4 + 12\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	3														6
2	4	12	9													25
3	4	24	45	27												100
4	4	36	117	162	81											400
5	4	48	225	513	567	243										1600
6	4	60	369	1188	2106	1944	729									6400
7	4	72	549	2295	5670	8262	6561	2187								25600
8	4	84	765	3942	12555	25272	31347	21870	6561							102400
9	4	96	1017	6237	24381	62937	107163	115911	72171	19683						409600
10	4	108	1305	9288	43092	136080	295974	437400	419904	236196	59049					1638400
11	4	120	1629	13203	70956	265356	704214	1325322	1732104	1495908	767637	177147				6553600
12	4	132	1989	18090	110565	478224	1500282	3437964	5708070	6692220	5255361	2480058	531441			26214400
13	4	144	2385	24057	164835	809919	2934954	7938810	16021962	23816430	25332021	18246141	7971615	1594323		104857600
14	4	156	2817	31212	237006	1304424	5364711	16743672	39838392	71882316	96781311	94242204	62710038	25509168	4782969	419430400

Technical details

Node id: n-33cd81904e6745b3 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9ab01a3eed27d5b9

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: 3,  3;
  2: 4, 12,    9;
  3: 4, 24,   45,   27;
  4: 4, 36,  117,  162,    81;
  5: 4, 48,  225,  513,   567,   243;
  6: 4, 60,  369, 1188,  2106,  1944,    729;
  7: 4, 72,  549, 2295,  5670,  8262,   6561,   2187;
  8: 4, 84,  765, 3942, 12555, 25272,  31347,  21870,  6561;
  9: 4, 96, 1017, 6237, 24381, 62937, 107163, 115911, 72171, 19683;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 4 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 4 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (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] = 3 + 3*x; v[3] = 4 + 12*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (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 Triangle read by rows. A130713 in the columns.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.