A008306 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\)

\(P_{2}(x) = 6 + 6\,x\)

\(P_{3}(x) = 24 + 40\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	6	6							12
3	24	40							64
4	120	260	45						425
5	720	1848	630						3198
6	5040	14616	7140	420					27216
7	40320	128448	79296	10080					258144
8	362880	1246752	910980	176400	4725				2701737
9	3628800	13273920	11036520	2821280	173250				30933770
10	39916800	153996480	141973128	44395120	4331250	62370			384675148
11	479001600	1935048960	1942608096	709325760	94294200	3243240			5163521856
12	6227020800	26193473280	28256836608	11682102432	1947245300	109729620	945945		74417353985
13	87178291200	380120670720	436231740672	199811451840	39674635000	3120357240	66216150		1146203362822
14	1307674368000	5888620684800	7132829258880	3562308509760	814158945600	81855774000	2913510600	16216200	18790377267840

Technical details

Node id: n-fe674caaab8df0fd · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-acb523b53b6d3c90

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:       2;
  2:       6,        6;
  3:      24,       40;
  4:     120,      260,       45;
  5:     720,     1848,      630;
  6:    5040,    14616,     7140,     420;
  7:   40320,   128448,    79296,   10080;
  8:  362880,  1246752,   910980,  176400,   4725;
  9: 3628800, 13273920, 11036520, 2821280, 173250;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n^3-n)*x^2*P(n-4,x) - (2*n^3-2*n)*x*P(n-3,x) - (n^3-n-(2*n^2+2*n)*x)*P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 6 + 6*x, P(3,x) = 24 + 40*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 6 + 6*x;
P[3, x_] := 24 + 40*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n)*x^2*P[n - 4, x] - (2*n^3 - 2*n)*x*P[n - 3, x] - (n^3 - n - (2*n^2 + 2*n)*x)*P[n - 2, x] + (2*n^2 + n - 1)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 6 + 6*x; v[4] = 24 + 40*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - n)*x^2*v[i - 4] - (2*n^3 - 2*n)*x*v[i - 3] - (n^3 - n - (2*n^2 + 2*n)*x)*v[i - 2] + (2*n^2 + n - 1)*v[i - 1])/(n)); 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 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows; related to number of preorders.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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.