A119308 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-32\,x^{2} + 16\,n x^{2}\right)\,P_{n-3}(x) + \left(2 + 4\,x - 4\,n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(7 + 12\,x + 5\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 9 + 28\,x + 16\,x^{2}\)

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	4														7
2	9	28	16													53
3	28	140	192	64												424
4	90	624	1408	1152	256											3530
5	297	2640	8400	11520	6400	1024										30281
6	1001	10868	44880	88000	83200	33792	4096									265837
7	3432	44044	224224	576576	788480	551936	172032	16384								2377108
8	11934	176800	1071616	3424512	6196736	6336512	3440640	851968	65536							21576254
9	41990	705432	4964544	19009536	43051008	58705920	46964736	20447232	4128768	262144						198281310
10	149226	2803640	22480800	100465920	274176000	470851584	505774080	326860800	116981760	19660800	1048576					1841253186
11	534888	11113784	100052480	511632000	1637222400	3405422592	4632477696	4044226560	2162688000	648806400	92274688	4194304				17250645792
12	1931540	43979680	439321344	2531327744	9304880640	22790135808	37721604096	41912893440	30441996288	13725859840	3506438144	427819008	16777216			162864964788
13	7020405	173838600	1908361520	12238237440	50856675584	143595319296	280947363840	381407330304	354163949568	217947045888	84119912448	18538823680	1962934272	67108864		1547933921709
14	25662825	686623140	8217452880	58066182720	269321212160	862275814400	1951206985728	3142203457536	3578498187264	2826901389312	1495715020800	500384661504	96183779328	8925478912	268435456	14798880343965

Technical details

Node id: n-8f99a549bab56bfb · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-debc431b0cc42a35

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:     3,      4;
  2:     9,     28,      16;
  3:    28,    140,     192,       64;
  4:    90,    624,    1408,     1152,      256;
  5:   297,   2640,    8400,    11520,     6400,     1024;
  6:  1001,  10868,   44880,    88000,    83200,    33792,     4096;
  7:  3432,  44044,  224224,   576576,   788480,   551936,   172032,    16384;
  8: 11934, 176800, 1071616,  3424512,  6196736,  6336512,  3440640,   851968,   65536;
  9: 41990, 705432, 4964544, 19009536, 43051008, 58705920, 46964736, 20447232, 4128768, 262144;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = (16*n-32)*x^2*P(n-3,x) - (4*n-2+(8*n-4)*x+16*n*x^2)*P(n-2,x) + (5*n+7+(8*n+12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 9 + 28*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 9 + 28*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 32)*x^2*P[n - 3, x] - (4*n - 2 + (8*n - 4)*x + 16*n*x^2)*P[n - 2, x] + (5*n + 7 + (8*n + 12)*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] = 3 + 4*x; v[3] = 9 + 28*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((16*n - 32)*x^2*v[i - 3] - (4*n - 2 + (8*n - 4)*x + 16*n*x^2)*v[i - 2] + (5*n + 7 + (8*n + 12)*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 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)

Proof chains ? JSON

Geometric basis map (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

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.