A200536 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 21\,x - 72\,x^{2} - 28\,x^{3} + 16\,x^{4} + 63\,n x + 96\,n x^{2} - 28\,n x^{3} - 64\,n x^{4} + 49\,n^{2} x^{2} + 112\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n}(x) = \left(54 + 378\,x + 1098\,x^{2} + 1694\,x^{3} + 1464\,x^{4} + 672\,x^{5} + 128\,x^{6} + 378\,n x + 2046\,n x^{2} + 4396\,n x^{3} + 4680\,n x^{4} + 2464\,n x^{5} + 512\,n x^{6} + 294\,n^{2} x^{2} + 1358\,n^{2} x^{3} + 2344\,n^{2} x^{4} + 1792\,n^{2} x^{5} + 512\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

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	56	72													134
2	36	672	2628	3584	1600											8520
3	216	6048	39528	108416	146400	96768	25088									422464
4	1296	48384	443232	1827840	4128400	5483520	4290048	1835008	331776							18389504
5	7776	362880	4276800	23546880	74604000	149030784	194965120	167444480	91238400	28672000	3964928					738114048
6	46656	2612736	37651392	259338240	1057752000	2822045184	5179445824	6689292288	6092651520	3842048000	1599848448	396361728	44302336			28023396352
7	279936	18289152	311568768	2576738304	12856233600	42924639744	101437865088	174981373952	223577335296	211973529600	147494330368	73293889536	24654249984	5035261952	471859200	1021607444480

All 15 rows

Technical details

Node id: n-51000fd75904f2d1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-95bcb04e021989f8

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:        6,        56,          72;
  2:       36,       672,        2628,         3584,          1600;
  3:      216,      6048,       39528,       108416,        146400,         96768,          25088;
  4:     1296,     48384,      443232,      1827840,       4128400,       5483520,        4290048,        1835008,          331776;
  5:     7776,    362880,     4276800,     23546880,      74604000,     149030784,      194965120,      167444480,        91238400,        28672000,         3964928;
  6:    46656,   2612736,    37651392,    259338240,    1057752000,    2822045184,     5179445824,     6689292288,      6092651520,      3842048000,      1599848448,       396361728,        44302336;
  7:   279936,  18289152,   311568768,   2576738304,   12856233600,   42924639744,   101437865088,   174981373952,    223577335296,    211973529600,    147494330368,     73293889536,     24654249984,      5035261952,      471859200;
  8:  1679616, 125411328,  2465676288,  23800283136,  140427302400,  563110797312,  1628165790720,  3515235827712,   5783379243264,   7323407974400,   7147684741120,   5339124596736,   3000220647424,   1228603916288,   346344652800,    60129542144,    4848615424;
  9: 10077696, 846526464, 18865446912, 208182804480, 1417987814400, 6638084296704, 22701283928064, 58872276320256, 118517522167296, 187840536934400, 236059591477248, 235489105281024, 185591146610688, 114222399750144, 53778500812800, 18715319992320, 4538304036864, 684913065984, 48452599808;

%F Formula

The row polynomials satisfy (9+(63*n-21)*x+(49*n^2+96*n-72)*x^2+(112*n^2-28*n-28)*x^3+(64*n^2-64*n+16)*x^4)*P(n,x) = (54+(378*n+378)*x+(294*n^2+2046*n+1098)*x^2+(1358*n^2+4396*n+1694)*x^3+(2344*n^2+4680*n+1464)*x^4+(1792*n^2+2464*n+672)*x^5+(512*n^2+512*n+128)*x^6)*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[((54 + (378*n + 378)*x + (294*n^2 + 2046*n + 1098)*x^2 + (1358*n^2 + 4396*n + 1694)*x^3 + (2344*n^2 + 4680*n + 1464)*x^4 + (1792*n^2 + 2464*n + 672)*x^5 + (512*n^2 + 512*n + 128)*x^6)*P[n - 1, x])/(9 + (63*n - 21)*x + (49*n^2 + 96*n - 72)*x^2 + (112*n^2 - 28*n - 28)*x^3 + (64*n^2 - 64*n + 16)*x^4)]];
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] = ((54 + (378*n + 378)*x + (294*n^2 + 2046*n + 1098)*x^2 + (1358*n^2 + 4396*n + 1694)*x^3 + (2344*n^2 + 4680*n + 1464)*x^4 + (1792*n^2 + 2464*n + 672)*x^5 + (512*n^2 + 512*n + 128)*x^6)*v[i - 1])/(9 + (63*n - 21)*x + (49*n^2 + 96*n - 72)*x^2 + (112*n^2 - 28*n - 28)*x^3 + (64*n^2 - 64*n + 16)*x^4)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) 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.