A119308 → reverse

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 1 + 5\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	5	3													9
3	1	9	14	4												28
4	1	14	40	30	5											90
5	1	20	90	125	55	6										297
6	1	27	175	385	315	91	7									1001
7	1	35	308	980	1274	686	140	8								3432
8	1	44	504	2184	4116	3528	1344	204	9							11934
9	1	54	780	4410	11340	14112	8568	2430	285	10						41990
10	1	65	1155	8250	27720	47124	41580	18810	4125	385	11					149226
11	1	77	1650	14520	61710	137214	165528	108900	38115	6655	506	12				534888
12	1	90	2288	24310	127413	358644	566280	509652	259545	72358	10296	650	13			1931540
13	1	104	3094	39039	247247	858858	1717716	2024451	1410981	572572	130130	15379	819	14		7020405
14	1	119	4095	60515	455455	1912911	4723719	7054905	6441435	3578575	1184183	223587	22295	1015	15	25662825

Technical details

Node id: n-4f9b456a67845b83 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-1c718f648220d69b

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: 1,  2;
  2: 1,  5,   3;
  3: 1,  9,  14,    4;
  4: 1, 14,  40,   30,     5;
  5: 1, 20,  90,  125,    55,     6;
  6: 1, 27, 175,  385,   315,    91,    7;
  7: 1, 35, 308,  980,  1274,   686,  140,    8;
  8: 1, 44, 504, 2184,  4116,  3528, 1344,  204,   9;
  9: 1, 54, 780, 4410, 11340, 14112, 8568, 2430, 285, 10;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = ((n-2)*x-(2*n-4)*x^2+(n-2)*x^3)*P(n-3,x) - (n-x+(3*n-1)*x^2)*P(n-2,x) + (2*n+3+(3*n+4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 5*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 5*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n - 2)*x - (2*n - 4)*x^2 + (n - 2)*x^3)*P[n - 3, x] - (n - x + (3*n - 1)*x^2)*P[n - 2, x] + (2*n + 3 + (3*n + 4)*x)*P[n - 1, x])/(n + 3)]];
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 + 2*x; v[3] = 1 + 5*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (((n - 2)*x - (2*n - 4)*x^2 + (n - 2)*x^3)*v[i - 3] - (n - x + (3*n - 1)*x^2)*v[i - 2] + (2*n + 3 + (3*n + 4)*x)*v[i - 1])/(n + 3)); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 for first differences of Catalan numbers.?
  - 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 ?

A119308: Triangle for first differences of Catalan numbers.

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.