A371259 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-144 - 60\,x + 9\,x^{2} - 168\,n - 32\,n x - 73\,n^{2} - 4\,n^{2} x - 14\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-231\,x + 69\,x^{2} + 4\,x^{3} - 182\,n x + 32\,n x^{2} - 47\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{3} x\right)\,P_{n-2}'(x) + \left(-75\,x^{2} + 41\,x^{3} - 40\,n x^{2} + 8\,n x^{3} - 5\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-6\,x^{3} + 4\,x^{4} - 2\,n x^{3}\right)\,P_{n-2}'''(x) + \left(34 + 4\,x + 16\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(17\,x - 4\,x^{2} + 4\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 36 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	36	4														40
2	1764	400	9													2173
3	112896	39456	1800	16												154168
4	9144576	4397184	306576	5504	25											13853865
5	914457600	568637568	54040608	1475200	13500	36										1538624512
6	110649369600	85541640192	10412297280	385211392	5342500	28656	49									206993889669
7	15933509222400	14899134670848	2233278957312	105183313920	1969572800	15974784	54880	64								33173091767008
8	2692763058585600	2983964432375808	536010818404608	30897202065408	733555467200	8012014848	41555136	97280	81							6244377120565969
9	527781559482777600	682163049518899200	143932689484425216	9882199518511104	285948849600000	3948657232896	27527296832	97176576	162324	100						1364049423136081848
10	118750850883624960000	176744866914769305600	43131414122416410624	3457712980858798080	118668471453273600	1991229236582400	17514099136576	82987483136	208858500	258000	121					342205522199655066637
11	30400217826207989760000	51554269941902829158400	14372254126221147979776	1324912455640988614656	52870065113901312000	1049014692225552384	11187108220784384	66775936634880	225333529344	419100000	393976	144				97705584683888992819944
12	8785662951774109040640000	16825184332481348606361600	5304380846935115786305536	555515656258566375604224	25385874427169523302400	583856004801185243136	7345805802101940992	53074672688021504	225346750643520	561501142400	794319504	581760	169			31496720916987494354106745
13	2846554796374811329167360000	6109034195832863283059097600	2159609003522129348886528000	254412414804579088594894848	13154756435654059240243200	345415485714951107174400	5025877037434553481216	42766192293983322112	219507036055796736	687540197801600	1302162873440	1434566016	834860	196		11383115651319177668443974224
14	1027606281491306889829416960000	2454999897518644558540308480000	966132897493882985245841817600	126974913660566731055727378432	7356385552593152164018176000	217902169039153683038404608	3611322003953057154969600	35474356815137040498688	214146591406748462592	810027855886080000	1927042615902976	2840790850560	2485923076	1168944	225	4583291924897352787861075073301

Technical details

Node id: n-2e9729dba03bdbf4 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5a3a97678ce8350c

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:                 36,                  4;
  2:               1764,                400,                  9;
  3:             112896,              39456,               1800,               16;
  4:            9144576,            4397184,             306576,             5504,              25;
  5:          914457600,          568637568,           54040608,          1475200,           13500,            36;
  6:       110649369600,        85541640192,        10412297280,        385211392,         5342500,         28656,          49;
  7:     15933509222400,     14899134670848,      2233278957312,     105183313920,      1969572800,      15974784,       54880,       64;
  8:   2692763058585600,   2983964432375808,    536010818404608,   30897202065408,    733555467200,    8012014848,    41555136,    97280,     81;
  9: 527781559482777600, 682163049518899200, 143932689484425216, 9882199518511104, 285948849600000, 3948657232896, 27527296832, 97176576, 162324, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4+14*n^3+73*n^2+168*n+144+(4*n^2+32*n+60)*x-9*x^2)*P(n-2,x) - ((4*n^3+47*n^2+182*n+231)*x-(4*n^2+32*n+69)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2+40*n+75)*x^2-(8*n+41)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n+6)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2+16*n+34+4*x)*P(n-1,x) + ((4*n+17)*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) = 36 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 36 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (4*n^2 + 32*n + 60)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 + 47*n^2 + 182*n + 231)*x - (4*n^2 + 32*n + 69)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 + 40*n + 75)*x^2 - (8*n + 41)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n + 6)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 + 16*n + 34 + 4*x)*P[n - 1, x] + ((4*n + 17)*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] = 36 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (4*n^2 + 32*n + 60)*x - 9*x^2)*v[i - 2] - ((4*n^3 + 47*n^2 + 182*n + 231)*x - (4*n^2 + 32*n + 69)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 + 40*n + 75)*x^2 - (8*n + 41)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n + 6)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 + 16*n + 34 + 4*x)*v[i - 1] + ((4*n + 17)*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, (2, 3)-Lah numbers.?
  - 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.