A228955 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 8\,x + 4\,x^{2} - 42\,n x + 12\,n x^{2} - 22\,n^{2} x + 13\,n^{2} x^{2} + 6\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(36\,n + 136\,n x - 104\,n x^{2} + 36\,n x^{3} - 42\,n^{2} x - 130\,n^{2} x^{2} + 60\,n^{2} x^{3} - 22\,n^{3} x - 25\,n^{3} x^{2} + 37\,n^{3} x^{3} + 6\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(72\,x - 216\,x^{2} + 216\,x^{3} - 72\,x^{4} - 172\,n x^{2} + 292\,n x^{3} - 120\,n x^{4} - 44\,n^{2} x^{2} + 118\,n^{2} x^{3} - 74\,n^{2} x^{4} + 20\,n^{3} x^{3} - 20\,n^{3} x^{4} + 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	2	24							26
3	6	144	54						204
4	24	960	1080						2064
5	120	7200	16200	1920					25440
6	720	60480	226800	80640					368640
7	5040	564480	3175200	2257920	126000				6128640
8	40320	5806080	45722880	54190080	9072000				114831360
9	362880	65318400	685843200	1219276800	408240000	13063680			2392104960
10	3628800	798336000	10777536000	26824089600	14968800000	1437004800			54809395200
11	39916800	10538035200	177829344000	590129971200	493970400000	94842316800	1955923200		1369305907200
12	479001600	149448499200	3082375296000	13151467929600	15411876480000	4931800473600	305124019200		37032571699200
13	6227020800	2266635571200	56099230387200	299195895398400	467493586560000	224396921548800	27766285747200	398529331200	1077623311564800
14	87178291200	36614882304000	1070985307392000	6981237559296000	14024807596800000	9424670705049600	1943640002304000	83691159552000	33565734390988800

Technical details

Node id: n-6ec34513293cdda0 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-1f0181c04d99414a

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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:      1,        4;
  2:      2,       24;
  3:      6,      144,        54;
  4:     24,      960,      1080;
  5:    120,     7200,     16200,       1920;
  6:    720,    60480,    226800,      80640;
  7:   5040,   564480,   3175200,    2257920,    126000;
  8:  40320,  5806080,  45722880,   54190080,   9072000;
  9: 362880, 65318400, 685843200, 1219276800, 408240000, 13063680;

%F Formula

The row polynomials satisfy (36-(22*n^2+42*n+8)*x+(n^4+6*n^3+13*n^2+12*n+4)*x^2)*P(n,x) = (36*n-(22*n^3+42*n^2-136*n)*x+(n^5+6*n^4-25*n^3-130*n^2-104*n)*x^2+(n^5+10*n^4+37*n^3+60*n^2+36*n)*x^3)*P(n-1,x) + (72*x-(44*n^2+172*n+216)*x^2+(2*n^4+20*n^3+118*n^2+292*n+216)*x^3-(2*n^4+20*n^3+74*n^2+120*n+72)*x^4)*d/dx 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[((36*n - (22*n^3 + 42*n^2 - 136*n)*x + (n^5 + 6*n^4 - 25*n^3 - 130*n^2 - 104*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*P[n - 1, x] + (72*x - (44*n^2 + 172*n + 216)*x^2 + (2*n^4 + 20*n^3 + 118*n^2 + 292*n + 216)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*D[P[n - 1, x], x])/(36 - (22*n^2 + 42*n + 8)*x + (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*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] = ((36*n - (22*n^3 + 42*n^2 - 136*n)*x + (n^5 + 6*n^4 - 25*n^3 - 130*n^2 - 104*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*v[i - 1] + (72*x - (44*n^2 + 172*n + 216)*x^2 + (2*n^4 + 20*n^3 + 118*n^2 + 292*n + 216)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*deriv(v[i - 1], x))/(36 - (22*n^2 + 42*n + 8)*x + (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^2)); 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 Table: T(n,k) = n!*binomial(n+1,2*k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.