A371081 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 12\,x + 9\,x^{2} - 12\,n - 16\,n x - 13\,n^{2} - 4\,n^{2} x - 6\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-23\,x + 21\,x^{2} + 4\,x^{3} - 42\,n x + 16\,n x^{2} - 23\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{3} x\right)\,P_{n-2}'(x) + \left(-15\,x^{2} + 25\,x^{3} - 20\,n x^{2} + 8\,n x^{3} - 5\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-2\,x^{3} + 4\,x^{4} - 2\,n x^{3}\right)\,P_{n-2}'''(x) + \left(10 + 4\,x + 8\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(9\,x - 4\,x^{2} + 4\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 16 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	16	4														20
2	400	208	9													617
3	14400	11792	1044	16												27252
4	705600	812288	111096	3456	25											1632465
5	45158400	68617728	12937248	615680	9000	36										127338092
6	3657830400	7042406400	1719796896	111657472	2483000	20016	49									12534194233
7	365783040000	866762496000	263496167424	21927529472	661132800	8079120	39788	64								1518638484668
8	44259747840000	126276931584000	46481067910656	4766233407488	183016644800	3020285952	22495312	72704	81							221970040240993
9	6373403688960000	21517840429056000	9394412406552576	1155035526303744	54299500768000	1136406608640	11399425856	55627776	124416	100						38496139413427108
10	1077105223434240000	4242990338850816000	2162157932439705600	312390272345407488	17497298731801600	446387022305280	5661968395776	37140156416	125271360	202000	121					7812593152058301641
11	211112623793111040000	958981247135170560000	563059158966978969600	94124625032160018432	6157234589643417600	186341825226000384	2872369694225920	23774033006592	107636848704	261514000	314116	144				1833624127593915915492
12	47500340353449984000000	246343649761776107520000	164881804747511055974400	31497372570805586362368	2369831413474023782400	83403147899486674944	1520346963933464576	15258298514587648	87028903488384	283517062400	512754440	471168	169			492677937687491626142897
13	12160087130483195904000000	71383316142567095009280000	53975977950157578377625600	11663674706500836331880448	997147210031493241651200	40193345459076217896960	849368881851431495680	10057399214048329728	69439932466834176	284641254726400	689677646944	953701632	685464	196		150221255981205604915754428
14	3514265180709643616256000000	23176834778713671566622720000	19645940501327661758093721600	4761427176733948005423513600	457966411352796076336742400	20889471184638239730106368	504023747596483224463360	6902441711342687223808	56128853672051277312	278145799882496000	847190922841120	1568475477504	1695750056	971376	225	51577834502619515321403304729

Technical details

Node id: n-b5c009c5db7508b7 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-580bcf0aafe7c36c

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:               16,                 4;
  2:              400,               208,                9;
  3:            14400,             11792,             1044,               16;
  4:           705600,            812288,           111096,             3456,             25;
  5:         45158400,          68617728,         12937248,           615680,           9000,            36;
  6:       3657830400,        7042406400,       1719796896,        111657472,        2483000,         20016,          49;
  7:     365783040000,      866762496000,     263496167424,      21927529472,      661132800,       8079120,       39788,       64;
  8:   44259747840000,   126276931584000,   46481067910656,    4766233407488,   183016644800,    3020285952,    22495312,    72704,     81;
  9: 6373403688960000, 21517840429056000, 9394412406552576, 1155035526303744, 54299500768000, 1136406608640, 11399425856, 55627776, 124416, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4+6*n^3+13*n^2+12*n+4+(4*n^2+16*n+12)*x-9*x^2)*P(n-2,x) - ((4*n^3+23*n^2+42*n+23)*x-(4*n^2+16*n+21)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2+20*n+15)*x^2-(8*n+25)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n+2)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2+8*n+10+4*x)*P(n-1,x) + ((4*n+9)*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) = 16 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 16 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (4*n^2 + 16*n + 12)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 + 23*n^2 + 42*n + 23)*x - (4*n^2 + 16*n + 21)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 + 20*n + 15)*x^2 - (8*n + 25)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n + 2)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 + 8*n + 10 + 4*x)*P[n - 1, x] + ((4*n + 9)*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] = 16 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (4*n^2 + 16*n + 12)*x - 9*x^2)*v[i - 2] - ((4*n^3 + 23*n^2 + 42*n + 23)*x - (4*n^2 + 16*n + 21)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 + 20*n + 15)*x^2 - (8*n + 25)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n + 2)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 + 8*n + 10 + 4*x)*v[i - 1] + ((4*n + 9)*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, 2)-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.