A241171 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	1														7
2	90	30	1													121
3	2520	1260	126	1												3907
4	113400	75600	13230	510	1											202741
5	7484400	6237000	1580040	126720	2046	1										15430207
6	681080400	681080400	227026800	28828800	1171170	8190	1									1619195761
7	81729648000	95351256000	39502663200	6972966000	494053560	10663380	32766	1								224061282907
8	12504636144000	16672848192000	8266953895200	1882311631200	196556560200	8203431600	96461910	131070	1							39531606447181
9	2375880867360000	3563821301040000	2059096751712000	576371099304000	80735951696400	5283068427000	133955839080	870123240	524286	1						8661323866026007
10	548828480360160000	914714133933600000	603711328396176000	200946724026048000	35589155470268400	3255176777252400	138104723433600	2167656876000	7838973450	2097150	1					2307185279184885001
11	151476660579404160000	277707211062240960000	206176565788633440000	79645466376075600000	17131288994099280000	2040089704773120000	126301275727704000	3550162645260000	34902001272600	70582218300	8388606	1				734307168916191403507
12	49229914688306352000000	98459829376612704000000	81229359235705480800000	35750295190317708000000	9072369992559313440000	1341213260195128320000	111457811625103944000	4781156660784504000	90324656188767000	560408322474000	635365793790	33554430	1			275199311597682485597221
13	18608907752179801056000000	40319300129722902288000000	36597518579286942076800000	18087070656483753724800000	5298779817618040353600000	940067598207159021600000	99212586047500685760000	5896518025761483120000	178083067096700766000	2283334779230505000	8984323401810120	5718795460560	134217726	1		119956934012963778952439407
14	8094874872198213459360000000	18888041368462498071840000000	18708155260191426661632000000	10262074176606835687104000000	3412553687304301522382400000	708490825421164249464000000	91325001120047485670880000	7057187670131421863040000	305134942368112442586000	6561690510910440906000	57487664033844080400	143909300701857600	51471172410930	536870910	1	60172884133239122388455031841

Technical details

Node id: n-277dc4a4d0ef7c4f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-18f46c82cbc91532

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:                6,                1;
  2:               90,               30,                1;
  3:             2520,             1260,              126,               1;
  4:           113400,            75600,            13230,             510,              1;
  5:          7484400,          6237000,          1580040,          126720,           2046,             1;
  6:        681080400,        681080400,        227026800,        28828800,        1171170,          8190,            1;
  7:      81729648000,      95351256000,      39502663200,      6972966000,      494053560,      10663380,        32766,         1;
  8:   12504636144000,   16672848192000,    8266953895200,   1882311631200,   196556560200,    8203431600,     96461910,    131070,      1;
  9: 2375880867360000, 3563821301040000, 2059096751712000, 576371099304000, 80735951696400, 5283068427000, 133955839080, 870123240, 524286, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2*n^2+3*n+1+n^2*x)*P(n-1,x) - ((4*n+1)*x+(2*n-1)*x^2)*d/dx P(n-1,x) + (2*x^2+x^3)*d^2/dx^2 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[(2*n^2 + 3*n + 1 + n^2*x)*P[n - 1, x] - ((4*n + 1)*x + (2*n - 1)*x^2)*D[P[n - 1, x], x] + (2*x^2 + x^3)*D[P[n - 1, x], {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] = (2*n^2 + 3*n + 1 + n^2*x)*v[i - 1] - ((4*n + 1)*x + (2*n - 1)*x^2)*deriv(v[i - 1], x) + (2*x^2 + x^3)*deriv(deriv(v[i - 1], x), x)); 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 read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A241171: Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

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.