A144445 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-43 + 42\,n\right)\,P_{n}(x) = \left(49\,x^{2} - 2058\,n x^{2}\right)\,P_{n-4}(x) + \left(-84\,x + 161\,x^{2} + 3528\,n x + 7742\,n x^{2} - 4116\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(49\,x^{2} - 49\,x^{3} - 2058\,n x^{2} + 2058\,n x^{3}\right)\,P_{n-3}'(x) + \left(-6 - 446\,x - 559\,x^{2} + 252\,n - 6048\,n x - 3066\,n x^{2} + 3528\,n^{2} x + 5635\,n^{2} x^{2} - 2058\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-42\,x + 343\,x^{2} - 301\,x^{3} + 1764\,n x + 49\,n x^{2} - 1813\,n x^{3} - 2058\,n^{2} x^{2} + 2058\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-37 + 516\,x - 210\,n - 1106\,n x + 588\,n^{2} x\right)\,P_{n-1}(x) + \left(-301\,x + 301\,x^{2} + 294\,n x - 294\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 1 + 46\,x + 3\,x^{2}\)

\(P_{3}(x) = 1 + 412\,x + 618\,x^{2} + 4\,x^{3}\)

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	46	3													50
3	1	412	618	4												1035
4	1	3354	19023	6708	5											29091
5	1	26904	400353	533804	67260	6										1028328
6	1	215318	7210836	23664924	12018060	645954	7									43755100
7	1	1722644	120072204	803183948	1003979935	240144408	6029254	8								2175132402
8	1	13781266	1914454437	23500813784	58437904315	35251220676	4467060353	55125064	9							123640359905
9	1	110250256	29768499267	627900631540	2759263704960	3311116445952	1098826105195	79382664712	496126152	10						7906864428045
10	1	882002190	456098589186	15816248753092	113973938509725	238891262504790	159563513913615	31632497506184	1368295767558	4410010950	11					561707147557302
11	1	7056017676	6927308675718	382898804672652	4299253894376250	14504318492395914	16921704907795233	6878806231002000	861522310513467	23091028919060	38808097218	12				43878568842465201
12	1	56448141578	104670357041259	9019160963292016	152098294946427260	781068080472309594	1457217567380845823	1041424107296412792	273776930903569068	22547902408230040	383791309151283	338688849468	13			3737640901174270195
13	1	451585132808	1576733876369109	208394989873303572	5135733795050637525	38558088004051530468	108341112025948996670	123818413743941710480	57837132006077295702	10271467590101275050	573086222151584823	6306935505476436	2935303363252	14		344751315433466675910
14	1	3612681062662	23709177955456224	4749647023687451004	167510253066152985380	1783793149283294047752	7229159392042424161283	12393499114933217814584	9294633504054545350221	2972988582138823412920	368522556745536567836	14248941071062353012	102739771140310304	25288767438634	15	34229231618209288411832

Technical details

Node id: n-4301181d23f6be2c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9e2530a6c09e04bb

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,        46,           3;
  3: 1,       412,         618,            4;
  4: 1,      3354,       19023,         6708,             5;
  5: 1,     26904,      400353,       533804,         67260,             6;
  6: 1,    215318,     7210836,     23664924,      12018060,        645954,             7;
  7: 1,   1722644,   120072204,    803183948,    1003979935,     240144408,       6029254,           8;
  8: 1,  13781266,  1914454437,  23500813784,   58437904315,   35251220676,    4467060353,    55125064,         9;
  9: 1, 110250256, 29768499267, 627900631540, 2759263704960, 3311116445952, 1098826105195, 79382664712, 496126152, 10;

%F Formula

The row polynomials satisfy (42*n-43)*P(n,x) = -(2058*n-49)*x^2*P(n-4,x) + ((3528*n-84)*x-(4116*n^2-7742*n-161)*x^2)*P(n-3,x) - ((2058*n-49)*x^2-(2058*n-49)*x^3)*d/dx P(n-3,x) + (252*n-6+(3528*n^2-6048*n-446)*x-(2058*n^3-5635*n^2+3066*n+559)*x^2)*P(n-2,x) + ((1764*n-42)*x-(2058*n^2-49*n-343)*x^2+(2058*n^2-1813*n-301)*x^3)*d/dx P(n-2,x) - (210*n+37-(588*n^2-1106*n+516)*x)*P(n-1,x) + ((294*n-301)*x-(294*n-301)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 46*x + 3*x^2, P(3,x) = 1 + 412*x + 618*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 46*x + 3*x^2;
P[3, x_] := 1 + 412*x + 618*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2058*n - 49)*x^2*P[n - 4, x] + ((3528*n - 84)*x - (4116*n^2 - 7742*n - 161)*x^2)*P[n - 3, x] - ((2058*n - 49)*x^2 - (2058*n - 49)*x^3)*D[P[n - 3, x], x] + (252*n - 6 + (3528*n^2 - 6048*n - 446)*x - (2058*n^3 - 5635*n^2 + 3066*n + 559)*x^2)*P[n - 2, x] + ((1764*n - 42)*x - (2058*n^2 - 49*n - 343)*x^2 + (2058*n^2 - 1813*n - 301)*x^3)*D[P[n - 2, x], x] - (210*n + 37 - (588*n^2 - 1106*n + 516)*x)*P[n - 1, x] + ((294*n - 301)*x - (294*n - 301)*x^2)*D[P[n - 1, x], x])/(42*n - 43)]];
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 + 46*x + 3*x^2; v[4] = 1 + 412*x + 618*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(2058*n - 49)*x^2*v[i - 4] + ((3528*n - 84)*x - (4116*n^2 - 7742*n - 161)*x^2)*v[i - 3] - ((2058*n - 49)*x^2 - (2058*n - 49)*x^3)*deriv(v[i - 3], x) + (252*n - 6 + (3528*n^2 - 6048*n - 446)*x - (2058*n^3 - 5635*n^2 + 3066*n + 559)*x^2)*v[i - 2] + ((1764*n - 42)*x - (2058*n^2 - 49*n - 343)*x^2 + (2058*n^2 - 1813*n - 301)*x^3)*deriv(v[i - 2], x) - (210*n + 37 - (588*n^2 - 1106*n + 516)*x)*v[i - 1] + ((294*n - 301)*x - (294*n - 301)*x^2)*deriv(v[i - 1], x))/(42*n - 43)); 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, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
  - 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, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
    
    ↑ 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.