A166960 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-7\,x^{2} - 2\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(7\,x^{2} - 13\,x^{3} + 9\,n x^{2} - 14\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(13\,x^{3} - 3\,x^{4} + 17\,n x^{3} - 4\,n x^{4}\right)\,P_{n-2}''(x) + \left(3\,x^{4} + 4\,n x^{4}\right)\,P_{n-2}'''(x) + \left(1 + 4\,n x\right)\,P_{n-1}(x) + \left(3\,x - 4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

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

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	24	9													34
3	1	108	189	16												314
4	1	448	2430	928	25											3832
5	1	1812	25902	27808	3525	36										59084
6	1	7272	253503	629120	218475	11448	49									1119868
7	1	29116	2379699	12319280	9393875	1355940	33565	64								25511540
8	1	116496	21875868	222491936	331091250	102922560	7181440	91776	81							685771408
9	1	466020	198979740	3832104000	10363143150	6089069160	912246720	34013184	238869	100						21430260944
10	1	1864120	1800254565	64143598080	300992216250	308744046576	86139587730	6942867456	148492197	599800	121					768913526896
11	1	7456524	16244233785	1055101642320	8326600382250	14148787216536	6742249512474	1006887763584	47176134453	609951100	1466157	144				31343665759328
12	1	29826144	146382653034	17170412655520	223002376401375	605278776198816	465176726535444	117278037538560	10192978519623	293963675200	2391527490	3510432	169			1438542079041808
13	1	119304628	1318249183194	277589196592096	5843347107776875	24680146741319196	29384472941068308	11758838759363328	1716211357647903	92315988011500	1712159014258	9043848288	8268325	196		73755961671398096
14	1	477218568	11867732309115	4469549794715008	150854742010848525	972629481039478056	1742170971693507243	1059603438677354496	243189082103214627	21944275524467800	765682968194793	9452757064320	33239229439	19217016	225	4195648578036819232

Technical details

Node id: n-c48848a8d81350db · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-935fde2140e0013c

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,     24,         9;
  3: 1,    108,       189,         16;
  4: 1,    448,      2430,        928,          25;
  5: 1,   1812,     25902,      27808,        3525,         36;
  6: 1,   7272,    253503,     629120,      218475,      11448,        49;
  7: 1,  29116,   2379699,   12319280,     9393875,    1355940,     33565,       64;
  8: 1, 116496,  21875868,  222491936,   331091250,  102922560,   7181440,    91776,     81;
  9: 1, 466020, 198979740, 3832104000, 10363143150, 6089069160, 912246720, 34013184, 238869, 100;

%F Formula

The row polynomials satisfy P(n,x) = (9*n^2-2*n-7)*x^2*P(n-2,x) + ((9*n+7)*x^2+(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) + ((17*n+13)*x^3-(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (4*n+3)*x^4*d^3/dx^3 P(n-2,x) + (1+4*n*x)*P(n-1,x) + (3*x-(4*n+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*n^2 - 2*n - 7)*x^2*P[n - 2, x] + ((9*n + 7)*x^2 + (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] + ((17*n + 13)*x^3 - (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (4*n + 3)*x^4*D[P[n - 2, x], {x, 3}] + (1 + 4*n*x)*P[n - 1, x] + (3*x - (4*n + 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*n^2 - 2*n - 7)*x^2*v[i - 2] + ((9*n + 7)*x^2 + (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) + ((17*n + 13)*x^3 - (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (4*n + 3)*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*n*x)*v[i - 1] + (3*x - (4*n + 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 T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.