A200536 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - x + 2\,x^{2} + 3\,n + 4\,n x\right)\,P_{n}(x) = \left(1 + 6\,x + 13\,x^{2} + 12\,x^{3} + 4\,x^{4} + 3\,n + 13\,n x + 18\,n x^{2} + 8\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	4	7	2																											13
2	7	32	49	28	4																									120
3	10	75	222	327	246	84	8																							972
4	13	136	602	1464	2121	1848	920	224	16																					7344
5	16	215	1270	4330	9400	13503	12890	7960	2960	560	32																			53136
6	19	312	2307	10140	29445	59352	84841	86196	61260	29280	8688	1344	64																	373248
7	22	427	3794	20433	74424	193641	370426	527887	561134	440412	249144	97104	23968	3136	128															2566080
8	25	560	5812	37072	162526	518784	1245748	2292176	3259921	3586288	3030832	1936704	909664	299264	63232	7168	256													17356032
9	28	711	8442	62244	319176	1207962	3494148	7891380	14090508	20011383	22616370	20241072	14187936	7642656	3068352	873216	161280	16128	512											115893504
10	31	880	11765	98460	578250	2532096	8573370	22970040	49422675	86163120	122246433	141194700	132281940	99727680	59696160	27787392	9740160	2442240	400640	35840	1024									765904896
11	34	1067	15862	148555	983290	4891194	18980412	58869822	148324110	306838455	524627862	743781591	874572006	850579620	680160360	442805088	231236544	94748544	29469440	6603520	974336	78848	2048							5018692608
12	37	1272	20814	215688	1588719	8850072	38729372	136526544	394393923	944782344	1891929006	3182938056	4510043097	5383157544	5401036872	4534304544	3161570544	1810752768	838466816	306797568	85755648	17369088	2328576	172032	4096					32651735040
13	40	1495	26702	303342	2461056	15178449	73957598	292098092	951948712	2592613881	5953896234	11600762238	19252648896	27268265391	32964226698	33956167752	29697720624	21925270224	13549394144	6926196992	2880537088	952064256	241632768	44648448	5484544	372736	8192			211147886592
14	43	1736	33607	415324	3680131	24895416	133675451	584774476	2122725605	6479168696	16790290517	37199841588	70807743825	116151584808	164455126233	200996914020	211770414492	191790327456	148622089616	97896390592	54313907648	25067254784	9460994816	2851150848	662631424	112566272	12759040	802816	16384	1358312177664

Technical details

Node id: n-a7af5531257fc90d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d31e0778cbd986bd

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:  4,   7,    2;
  2:  7,  32,   49,    28,      4;
  3: 10,  75,  222,   327,    246,      84,       8;
  4: 13, 136,  602,  1464,   2121,    1848,     920,     224,       16;
  5: 16, 215, 1270,  4330,   9400,   13503,   12890,    7960,     2960,      560,       32;
  6: 19, 312, 2307, 10140,  29445,   59352,   84841,   86196,    61260,    29280,     8688,     1344,       64;
  7: 22, 427, 3794, 20433,  74424,  193641,  370426,  527887,   561134,   440412,   249144,    97104,    23968,    3136,     128;
  8: 25, 560, 5812, 37072, 162526,  518784, 1245748, 2292176,  3259921,  3586288,  3030832,  1936704,   909664,  299264,   63232,   7168,    256;
  9: 28, 711, 8442, 62244, 319176, 1207962, 3494148, 7891380, 14090508, 20011383, 22616370, 20241072, 14187936, 7642656, 3068352, 873216, 161280, 16128, 512;

%F Formula

The row polynomials satisfy (3*n-2+(4*n-1)*x+2*x^2)*P(n,x) = (3*n+1+(13*n+6)*x+(18*n+13)*x^2+(8*n+12)*x^3+4*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 1 + (13*n + 6)*x + (18*n + 13)*x^2 + (8*n + 12)*x^3 + 4*x^4)*P[n - 1, x])/(3*n - 2 + (4*n - 1)*x + 2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + 1 + (13*n + 6)*x + (18*n + 13)*x^2 + (8*n + 12)*x^3 + 4*x^4)*v[i - 1])/(3*n - 2 + (4*n - 1)*x + 2*x^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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Parent-row provers of polytool polytool ? 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.