A123160 → reverse

nonnegative coefficients

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

Recurrence

\(\left(5 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-9\,x^{2} + 9\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(30 + 14\,x - 32\,n - 16\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)

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

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

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	1														2
3	3	4	1													8
4	10	18	9	1												38
5	35	80	60	16	1											192
6	126	350	350	150	25	1										1002
7	462	1512	1890	1120	315	36	1									5336
8	1716	6468	9702	7350	2940	588	49	1								28814
9	6435	27456	48048	44352	23100	6720	1008	64	1							157184
10	24310	115830	231660	252252	162162	62370	13860	1620	81	1						864146
11	92378	486200	1093950	1372800	1051050	504504	150150	26400	2475	100	1					4780008
12	352716	2032316	5080790	7220070	6417840	3699696	1387386	330330	47190	3630	121	1				26572086
13	1352078	8465184	23279256	36951200	37413090	25204608	11435424	3459456	675675	80080	5148	144	1			148321344
14	5200300	35154028	105462084	184940756	210159950	162123390	86465808	31855824	7963956	1301300	130130	7098	169	1		830764794
15	20058300	145608400	473227300	908596416	1145210066	995834840	611080470	266048640	81477396	17153136	2382380	203840	9555	196	1	4666890936

Technical details

Node id: n-d3ec0f3c81ea7de7 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-82e5c201b4913233

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,      1;
   3:     3,      4,      1;
   4:    10,     18,      9,      1;
   5:    35,     80,     60,     16,      1;
   6:   126,    350,    350,    150,     25,     1;
   7:   462,   1512,   1890,   1120,    315,    36,     1;
   8:  1716,   6468,   9702,   7350,   2940,   588,    49,    1;
   9:  6435,  27456,  48048,  44352,  23100,  6720,  1008,   64,  1;
  10: 24310, 115830, 231660, 252252, 162162, 62370, 13860, 1620, 81, 1;

%F Formula

The row polynomials satisfy (2*n^2-7*n+5)*P(n,x) = -(2*n^2-9*n+9)*x^2*P(n-2,x) + (8*n^2-32*n+30+(4*n^2-16*n+14)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9)*x^2*P[n - 2, x] + (8*n^2 - 32*n + 30 + (4*n^2 - 16*n + 14)*x)*P[n - 1, x])/(2*n^2 - 7*n + 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9)*x^2*v[i - 2] + (8*n^2 - 32*n + 30 + (4*n^2 - 16*n + 14)*x)*v[i - 1])/(2*n^2 - 7*n + 5)); 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) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.?
  - 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 ?

A123160: Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

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.