A299989 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 3\,x + 3\,x^{2} + x^{3} + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(9 + 33\,x + 46\,x^{2} + 30\,x^{3} + 9\,x^{4} + x^{5} + 12\,n x + 22\,n x^{2} + 12\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	x²⁸	x²⁹	P_n(1)
0	1	1																													2
1	3	11	7	1																											22
2	9	57	90	54	13	1																									224
3	27	243	621	715	421	129	19	1																							2176
4	81	945	3348	5772	5630	3286	1156	236	25	1																					20480
5	243	3483	15795	36405	50070	44094	25614	9850	2455	375	31	1																			188416
6	729	12393	68526	198018	352755	418527	343924	199228	81519	23255	4478	546	37	1																	1703936
7	2187	43011	280665	975807	2137779	3199959	3419913	2673395	1545961	662137	208243	47117	7385	749	43	1															15204352
8	6561	146529	1102248	4480920	11659788	21103740	27841464	27524712	20720310	11968614	5308888	1797544	458220	85820	11336	984	49	1													134217728
9	19683	492075	4192479	19514601	58830300	125000172	196501788	235471860	219151818	160187338	92459170	42163614	15125772	4227132	905004	144516	16491	1251	55	1											1174405120
10	59049	1633689	15549570	81553230	279575145	682359309	1246144824	1756886760	1951611570	1730898210	1235644652	712841812	332452090	124940850	37566840	8929128	1646853	229125	23010	1550	61	1									10200547328
11	177147	5373459	56509893	329749299	1266961905	3493581597	7271797599	11799747333	15251491134	15933330270	13586031426	9512704062	5485925394	2606422522	1017700750	324803754	83978103	17354271	2810313	346335	31053	1881	67	1							88046829568
12	531441	17537553	201947580	1297739556	5524112682	16985444850	39702806748	72910512180	107567098551	129491617167	128586760632	106110548424	73105995820	42149453308	20340428376	8200150024	2750130031	762054975	172715532	31558164	4553802	503602	40780	2244	73	1					755914244096
13	1594323	56864187	711599499	4993478685	23327464914	79139792394	205288821738	421011308718	698439784185	952609352409	1080710907705	1028316576807	825404832876	560944700060	323358718492	158133779284	65505022037	22908296125	6727548685	1646196123	331988514	54350010	7071194	709150	52351	2639	79	1			6459630813184
14	4782969	183347145	2477577942	18850920858	95889375855	355814334603	1014415919244	2298425611716	4236481219113	6459522918849	8247477103866	8897465358054	8163367532775	6399033269979	4298065786344	2477343704760	1225471986067	519619715251	188353456746	58118562502	15171905749	3322517913	603386316	89457732	10595403	971971	65926	3066	85	1	54975581388800

Technical details

Node id: n-7d4fd94324eb099c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-610b16754d4e841f

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:     1,      1;
  1:     3,     11,       7,        1;
  2:     9,     57,      90,       54,       13,         1;
  3:    27,    243,     621,      715,      421,       129,        19,         1;
  4:    81,    945,    3348,     5772,     5630,      3286,      1156,       236,        25,         1;
  5:   243,   3483,   15795,    36405,    50070,     44094,     25614,      9850,      2455,       375,       31,        1;
  6:   729,  12393,   68526,   198018,   352755,    418527,    343924,    199228,     81519,     23255,     4478,      546,       37,       1;
  7:  2187,  43011,  280665,   975807,  2137779,   3199959,   3419913,   2673395,   1545961,    662137,   208243,    47117,     7385,     749,     43,      1;
  8:  6561, 146529, 1102248,  4480920, 11659788,  21103740,  27841464,  27524712,  20720310,  11968614,  5308888,  1797544,   458220,   85820,  11336,    984,    49,    1;
  9: 19683, 492075, 4192479, 19514601, 58830300, 125000172, 196501788, 235471860, 219151818, 160187338, 92459170, 42163614, 15125772, 4227132, 905004, 144516, 16491, 1251, 55, 1;

%F Formula

The row polynomials satisfy (3+(4*n+3)*x+(2*n+3)*x^2+x^3)*P(n,x) = (9+(12*n+33)*x+(22*n+46)*x^2+(12*n+30)*x^3+(2*n+9)*x^4+x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + x.

%t Mathematica

P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((9 + (12*n + 33)*x + (22*n + 46)*x^2 + (12*n + 30)*x^3 + (2*n + 9)*x^4 + x^5)*P[n - 1, x])/(3 + (4*n + 3)*x + (2*n + 3)*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (12*n + 33)*x + (22*n + 46)*x^2 + (12*n + 30)*x^3 + (2*n + 9)*x^4 + x^5)*v[i - 1])/(3 + (4*n + 3)*x + (2*n + 3)*x^2 + x^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 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: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…?
  - 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.