A269944 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 16\,x + 9\,x^{2} + 32\,n + 16\,n x - 24\,n^{2} - 4\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(16\,x^{2} + 4\,x^{3} - 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(5 + 4\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	4														4
2	0	4	9													13
3	0	16	45	16												77
4	0	144	441	224	25											834
5	0	2304	7380	4368	750	36										14838
6	0	57600	189684	122320	25575	1980	49									397208
7	0	2073600	6958224	4740736	1111825	108108	4459	64								14997016
8	0	101606400	345618576	244666240	61886825	6898320	365638	8960	81							761051040
9	0	6502809600	22348203264	16273072384	4343047800	530609508	32790212	1051008	16524	100						50031600400
10	0	526727577600	1824835785984	1357849002240	377213547400	49233358980	3378225669	127959680	2668626	28500	121					4139368154800
11	0	52672757760000	183668715648000	139029052732416	39842993806000	5466523406256	404834638845	17208344384	428811570	6144600	46585	144				421102521338800
12	0	6373403688960000	22342428298368000	17149037541774336	5038235145420400	718823243237616	56425537047649	2610973280384	73665510831	1272893600	13071751	72864	169			51681039379637600
13	0	917770131210240000	3231649833265152000	2509181278545936384	752301232099560000	110765605635622080	9103675638157100	449678813008960	13912346617650	274241753500	3422533400	26048880	109850	196		7531235625244740000
14	0	155103152174530560000	548213804617033728000	429796791333401296896	131059503972553665600	19802701126643497920	1689285479408146620	87886234517739840	2920311326097315	63522592906500	910240666335	8475358320	49135905	160524	225	1285756109690772960000

Technical details

Node id: n-ab6e8549e7f75941 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-a25633ab4d4ed031

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: 0,          4;
  2: 0,          4,           9;
  3: 0,         16,          45,          16;
  4: 0,        144,         441,         224,         25;
  5: 0,       2304,        7380,        4368,        750,        36;
  6: 0,      57600,      189684,      122320,      25575,      1980,       49;
  7: 0,    2073600,     6958224,     4740736,    1111825,    108108,     4459,      64;
  8: 0,  101606400,   345618576,   244666240,   61886825,   6898320,   365638,    8960,    81;
  9: 0, 6502809600, 22348203264, 16273072384, 4343047800, 530609508, 32790212, 1051008, 16524, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^2-16*n+16)*x-9*x^2)*P(n-2,x) + ((4*n^2-16*n+16)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^2 - 16*n + 16)*x - 9*x^2)*P[n - 2, x] + ((4*n^2 - 16*n + 16)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 5 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^2 - 16*n + 16)*x - 9*x^2)*v[i - 2] + ((4*n^2 - 16*n + 16)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 5 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool 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.