A036969 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-2}(x) + \left(9\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(17\,x^{3}\right)\,P_{n-2}''(x) + \left(4\,x^{4}\right)\,P_{n-2}'''(x) + \left(1 + 4\,x\right)\,P_{n-1}(x) + \left(3\,x - 4\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

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	4														5
2	1	20	9													30
3	1	84	126	16												227
4	1	340	1323	480	25											2169
5	1	1364	12672	10032	1375	36										25480
6	1	5460	117117	183040	50050	3276	49									358993
7	1	21844	1066338	3136848	1537250	190008	6860	64								5959213
8	1	87380	9646191	52085280	43332575	9053928	594762	13056	81							114813254
9	1	349524	87012324	850513264	1164697625	388340316	41466740	1612416	23085	100						2534015395
10	1	1398100	783897345	13762900800	30446367600	15657415956	2560444579	157355264	3910599	38500	121					63373728865
11	1	5592404	7058221830	221600008080	782663722500	607509743760	146773267200	13414991040	515911275	8677900	61226	144				1779550197360
12	1	22369620	63536579379	3558148079200	19912843075125	22997386535760	8018778355140	1050263285760	58767161310	1504717500	17908605	93600	169			55661268161169
13	1	89478484	571879546056	57043323186096	503380683251875	856580409315540	424222137742200	77690356711680	6089379537150	223023801000	3987649380	34791120	138411	196		1925805305149189
14	1	357913940	5147117241093	913709845726080	12673647273775150	31561762919242140	21952785862047285	5526268887006720	591566600222370	29820132615000	752364374190	9755553600	64222704	198940	225	73255471180139438

Technical details

Node id: n-f56f01f99b47548b · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3ac78d4b619e6815

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,      4;
  2: 1,     20,        9;
  3: 1,     84,      126,        16;
  4: 1,    340,     1323,       480,         25;
  5: 1,   1364,    12672,     10032,       1375,        36;
  6: 1,   5460,   117117,    183040,      50050,      3276,       49;
  7: 1,  21844,  1066338,   3136848,    1537250,    190008,     6860,      64;
  8: 1,  87380,  9646191,  52085280,   43332575,   9053928,   594762,   13056,    81;
  9: 1, 349524, 87012324, 850513264, 1164697625, 388340316, 41466740, 1612416, 23085, 100;

%F Formula

The row polynomials satisfy P(n,x) = 9*x^2*P(n-2,x) + (9*x^2+4*x^3)*d/dx P(n-2,x) + 17*x^3*d^2/dx^2 P(n-2,x) + 4*x^4*d^3/dx^3 P(n-2,x) + (1+4*x)*P(n-1,x) + (3*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 2, x] + (9*x^2 + 4*x^3)*D[P[n - 2, x], x] + 17*x^3*D[P[n - 2, x], {x, 2}] + 4*x^4*D[P[n - 2, x], {x, 3}] + (1 + 4*x)*P[n - 1, x] + (3*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {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; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = 9*x^2*v[i - 2] + (9*x^2 + 4*x^3)*deriv(v[i - 2], x) + 17*x^3*deriv(deriv(v[i - 2], x), x) + 4*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*x)*v[i - 1] + (3*x - 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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 read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - ↑ Iterated real-rootedness preservers (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.