A120986 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(4\,x + 2\,x^{2} + 2\,n - 26\,n x - 3\,n x^{2} + 4\,n^{2} + 22\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + x^{2} + x^{3} + 8\,n x - 7\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	1	6	5													12
4	1	12	28	14												55
5	1	20	90	120	42											273
6	1	30	220	550	495	132										1428
7	1	42	455	1820	3003	2002	429									7752
8	1	56	840	4900	12740	15288	8008	1430								43263
9	1	72	1428	11424	42840	79968	74256	31824	4862							246675
10	1	90	2280	23940	122094	325584	465120	348840	125970	16796						1430715
11	1	110	3465	46200	307230	1106028	2238390	2558160	1598850	497420	58786					8414640
12	1	132	5060	83490	701316	3272808	8883336	14276790	13483635	7191272	1961256	208012				50067108
13	1	156	7150	143000	1480050	8682960	30390360	65122200	85925125	68740100	31870410	7726160	742900			300830572
14	1	182	9828	234234	2927925	21081060	92355120	253976580	444459015	493843350	341200860	139582170	30421755	2674440		1822766520
15	1	210	13195	369460	5486481	47549502	254729475	873358200	1952925975	2864291430	2734096365	1657028100	605452575	119759850	9694845	11124755664

Technical details

Node id: n-849696e2738fcb3d · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-85c07d3149a3188a

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    5;
   4: 1, 12,   28,    14;
   5: 1, 20,   90,   120,     42;
   6: 1, 30,  220,   550,    495,    132;
   7: 1, 42,  455,  1820,   3003,   2002,    429;
   8: 1, 56,  840,  4900,  12740,  15288,   8008,   1430;
   9: 1, 72, 1428, 11424,  42840,  79968,  74256,  31824,   4862;
  10: 1, 90, 2280, 23940, 122094, 325584, 465120, 348840, 125970, 16796;

%F Formula

The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (4*n^2+2*n+(22*n^2-26*n+4)*x+(n^2-3*n+2)*x^2)*P(n-1,x) + ((8*n-2)*x-(7*n-1)*x^2-(n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (22*n^2 - 26*n + 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] + ((8*n - 2)*x - (7*n - 1)*x^2 - (n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 2*n + (22*n^2 - 26*n + 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] + ((8*n - 2)*x - (7*n - 1)*x^2 - (n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
  - 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

Related ?

Same fingerprint ?

A120986: Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k…

References

[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.