A200536 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

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	P_n(1)
0																													0
1	3	4																											7
2	6	26	36	16																									84
3	9	66	189	264	180	48																							756
4	12	124	540	1284	1800	1488	672	128																					6048
5	15	200	1170	3940	8415	11820	10920	6400	2160	320																			45360
6	18	294	2160	9420	27090	53934	75852	75360	51840	23520	6336	768																	326592
7	21	406	3591	19236	69615	179718	340473	479248	501228	384720	210672	77952	17472	1792															2286144
8	24	536	5544	35224	153720	488040	1164408	2125832	2994192	3253600	2705472	1690752	768768	240128	46080	4096													15676416
9	27	684	8100	59544	304290	1147104	3302964	7419528	13163067	18548820	20761488	18353664	12658464	6668928	2592000	700416	117504	9216											105815808
10	30	850	11340	94680	554580	2421180	8169840	21802920	46697310	80974530	114148980	130817520	121380480	90390720	53239680	24238080	8225280	1958400	291840	20480									705438720
11	33	1034	15345	143440	947430	4701708	18196794	56270280	141290325	291139530	495513909	698734872	816344100	787783920	623890080	401212416	206160768	82621440	24879360	5294080	709632	45056							4655895552
12	36	1236	20196	208956	1536480	8542776	37305576	131197176	377994276	902782980	1801650708	3019151628	4258447128	5055584688	5039923680	4198309632	2899176192	1640212992	747343872	267463680	72382464	13922304	1695744	98304					30474952704
13	39	1456	25974	294684	2387385	14700972	71507436	281882744	916713369	2490756840	5704820550	11082142188	18329137047	25858331772	31117203000	31881687552	27705115152	20295557568	12421863168	6272414720	2566916352	829830144	203913216	35782656	3993600	212992			198087192576
14	42	1694	32760	404404	3579030	24179610	129645516	566253688	2051943894	6251174930	16165173024	35730280404	67830220458	110933875542	156531277980	190561495488	199860321024	180033837984	138620209728	90600590080	49783878144	22696593920	8429331456	2484658176	559104000	90202112	9289728	458752	1279948013568

Technical details

Node id: n-a35b6e0a133d5096 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-da2813591e6631ab

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  3,   4;
  2:  6,  26,   36,    16;
  3:  9,  66,  189,   264,    180,      48;
  4: 12, 124,  540,  1284,   1800,    1488,     672,     128;
  5: 15, 200, 1170,  3940,   8415,   11820,   10920,    6400,     2160,      320;
  6: 18, 294, 2160,  9420,  27090,   53934,   75852,   75360,    51840,    23520,     6336,      768;
  7: 21, 406, 3591, 19236,  69615,  179718,  340473,  479248,   501228,   384720,   210672,    77952,    17472,    1792;
  8: 24, 536, 5544, 35224, 153720,  488040, 1164408, 2125832,  2994192,  3253600,  2705472,  1690752,   768768,  240128,   46080,   4096;
  9: 27, 684, 8100, 59544, 304290, 1147104, 3302964, 7419528, 13163067, 18548820, 20761488, 18353664, 12658464, 6668928, 2592000, 700416, 117504, 9216;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 3*n*x + 2*n*x^2)*v[i - 1])/(n - 1)); 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.