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

A241171 ↗ nonnegative coefficients

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

Recurrence from A241171

\(P_{n}(x) = \left(1 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

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

b-files: A241171 ?

Technical details

Node id: n-4ef34abfaa61f0d6 · revision 0 · JSON
Definition source: oeis_catalog:A241171
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 ?

A241171 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A094088

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,      6;
  2: 1,     30,        90;
  3: 1,    126,      1260,         2520;
  4: 1,    510,     13230,        75600,        113400;
  5: 1,   2046,    126720,      1580040,       6237000,        7484400;
  6: 1,   8190,   1171170,     28828800,     227026800,      681080400,       681080400;
  7: 1,  32766,  10663380,    494053560,    6972966000,    39502663200,     95351256000,      81729648000;
  8: 1, 131070,  96461910,   8203431600,  196556560200,  1882311631200,   8266953895200,   16672848192000,   12504636144000;
  9: 1, 524286, 870123240, 133955839080, 5283068427000, 80735951696400, 576371099304000, 2059096751712000, 3563821301040000, 2375880867360000;

%F Formula

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) + (3*x+9*x^2)*d/dx P(n-1,x) + (x^2+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[(1 + 6*x)*P[n - 1, x] + (3*x + 9*x^2)*D[P[n - 1, x], x] + (x^2 + 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] = (1 + 6*x)*v[i - 1] + (3*x + 9*x^2)*deriv(v[i - 1], x) + (x^2 + 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

A241171 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,      6;
   3: 1,     30,        90;
   4: 1,    126,      1260,         2520;
   5: 1,    510,     13230,        75600,        113400;
   6: 1,   2046,    126720,      1580040,       6237000,        7484400;
   7: 1,   8190,   1171170,     28828800,     227026800,      681080400,       681080400;
   8: 1,  32766,  10663380,    494053560,    6972966000,    39502663200,     95351256000,      81729648000;
   9: 1, 131070,  96461910,   8203431600,  196556560200,  1882311631200,   8266953895200,   16672848192000,   12504636144000;
  10: 1, 524286, 870123240, 133955839080, 5283068427000, 80735951696400, 576371099304000, 2059096751712000, 3563821301040000, 2375880867360000;

%F Formula

The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) + (3*x+9*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 6*x)*P[n - 1, x] + (3*x + 9*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + 6*x)*v[i - 1] + (3*x + 9*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers

Proof chains ? JSON

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.