A154233 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(x^{2} - n x^{2} - 5\,n^{2} x^{2} + n^{3} x^{2} + 3\,n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-4}(x) + \left(2\,x + 4\,n x + 2\,n x^{2} + 2\,n^{2} x^{2} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(-1 + x^{2} - n - 6\,n x - n x^{2} + 2\,n^{3} x\right)\,P_{n-2}(x) + \left(-2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 1 + 14\,x + 3\,x^{2}\)

\(P_{3}(x) = 1 + 38\,x + 57\,x^{2} + 4\,x^{3}\)

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	2														3
2	1	14	3													18
3	1	38	57	4												100
4	1	78	513	156	5											753
5	1	138	2283	3044	345	6										5817
6	1	222	7287	34132	12145	666	7									54460
7	1	334	19005	211268	264085	38010	1169	8								533880
8	1	478	43149	926444	3557385	1389666	100681	1912	9							6019725
9	1	658	88455	3239236	28219005	33862806	5668663	235880	2961	10						71317675
10	1	878	167595	9628164	158496045	533031570	221894463	19256328	502785	4390	11					942982230
11	1	1142	298209	25301764	700956145	5159254410	6019130145	1121529832	56928969	994030	6281	12				13084400940
12	1	1454	504057	60335676	2598040125	35480666154	108428135865	47307554872	4676472225	150839190	1848209	8724	13			198704406565
13	1	1818	816291	132975524	8397983825	190845988998	1239284802623	1416325488712	286268983497	16795967650	365682691	3265164	11817	14		3158421968625
14	1	2238	1274847	274527796	24326898585	852512032938	10113307553671	28731485715464	13002823997577	1420853388230	53519176887	823583388	5524337	15666	15	54199933691640

Technical details

Node id: n-39a77531463ce78d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5253ff5901def6be

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,  14,     3;
  3: 1,  38,    57,       4;
  4: 1,  78,   513,     156,        5;
  5: 1, 138,  2283,    3044,      345,        6;
  6: 1, 222,  7287,   34132,    12145,      666,       7;
  7: 1, 334, 19005,  211268,   264085,    38010,    1169,      8;
  8: 1, 478, 43149,  926444,  3557385,  1389666,  100681,   1912,    9;
  9: 1, 658, 88455, 3239236, 28219005, 33862806, 5668663, 235880, 2961, 10;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(n^5-3*n^4-n^3+5*n^2+n-1)*x^2*P(n-4,x) - ((2*n^3-4*n-2)*x+(2*n^3-2*n^2-2*n)*x^2)*P(n-3,x) - (n+1-(2*n^3-6*n)*x+(n-1)*x^2)*P(n-2,x) + (2*n+(2*n-2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 14*x + 3*x^2, P(3,x) = 1 + 38*x + 57*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 14*x + 3*x^2;
P[3, x_] := 1 + 38*x + 57*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*P[n - 4, x] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*P[n - 3, x] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*P[n - 2, x] + (2*n + (2*n - 2)*x)*P[n - 1, x])/(n - 1)]];
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] = 1 + 2*x; v[3] = 1 + 14*x + 3*x^2; v[4] = 1 + 38*x + 57*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*v[i - 4] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*v[i - 3] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*v[i - 2] + (2*n + (2*n - 2)*x)*v[i - 1])/(n - 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 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

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.