A143491 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 24\,x - 16\,x^{2} - 6\,n - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(7 + 8\,x + 2\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 12 + 56\,x + 48\,x^{2}\)

\(P_{1}(x) = 60 + 376\,x + 576\,x^{2} + 256\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	12	56	48								116
1	60	376	576	256							1268
2	360	2736	5712	4608	1280						14696
3	2520	22032	56400	62720	32000	6144					181816
4	20160	196416	583392	802560	569600	202752	28672				2403552
5	181440	1929024	6429024	10334464	9139200	4558848	1204224	131072			33907296
6	1814400	20741760	75864384	137632768	143064320	89456640	33316864	6815744	589824		509296704
7	19958400	242674560	958958784	1918570496	2261871360	1670731776	783949824	227278848	37158912	2621440	8123774400

All 15 rows

Technical details

Node id: n-ab41235953e80923 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-466d95fa7350938a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         12,          56,           48;
  1:         60,         376,          576,          256;
  2:        360,        2736,         5712,         4608,         1280;
  3:       2520,       22032,        56400,        62720,        32000,         6144;
  4:      20160,      196416,       583392,       802560,       569600,       202752,        28672;
  5:     181440,     1929024,      6429024,     10334464,      9139200,      4558848,      1204224,       131072;
  6:    1814400,    20741760,     75864384,    137632768,    143064320,     89456640,     33316864,      6815744,      589824;
  7:   19958400,   242674560,    958958784,   1918570496,   2261871360,   1670731776,    783949824,    227278848,    37158912,    2621440;
  8:  239500800,  3071761920,  12963552768,  28137292800,  36735308800,  30905763840,  17204146176,   6311116800,  1468661760,  196608000,   11534336;
  9: 3113510400, 41848911360, 186956757504, 434923754496, 618245478400, 578104412160, 367880798208, 160692043776, 47492628480, 9083289600, 1015021568, 50331648;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(8*n+24)*x+16*x^2)*P(n-2,x) + (2*n+7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 12 + 56*x + 48*x^2, P(1,x) = 60 + 376*x + 576*x^2 + 256*x^3.

%t Mathematica

P[0, x_] := 12 + 56*x + 48*x^2;
P[1, x_] := 60 + 376*x + 576*x^2 + 256*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (8*n + 24)*x + 16*x^2)*P[n - 2, x] + (2*n + 7 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 56*x + 48*x^2; v[2] = 60 + 376*x + 576*x^2 + 256*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (8*n + 24)*x + 16*x^2)*v[i - 2] + (2*n + 7 + 8*x)*v[i - 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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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_A013611 of Unsigned 2-Stirling numbers of the first kind.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A136124 → basis A013611 → θ+1

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.