A225469 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 - 12\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-24\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-16\,x^{2} + 16\,x^{3}\right)\,P_{n-2}''(x) + \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(8\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	4														7
3	9	40	9													58
4	27	316	189	16												548
5	81	2320	2790	576	25											5792
6	243	16564	35910	13600	1375	36										67728
7	729	116920	432279	267840	47375	2808	49									868000
8	2187	821356	5018139	4786096	1318625	132804	5145	64								12084416
9	6561	5758240	57047580	80712576	32532150	4953312	319676	8704	81							181338880
10	19683	40333924	640479420	1312106560	744224250	160772472	15373260	687360	13851	100						2914010880
11	59049	282416200	7136024949	20820228480	16190427250	4769449776	633907218	41387520	1354725	21000	121					49875276288
12	177147	1977149596	79131710889	324989693776	340149724750	133011560088	23607245970	2110973568	99796455	2491500	30613	144				905080554496
13	531441	13840755760	874897406370	5015524003776	6970641166775	3549081485664	818439153532	96274134528	6164576847	220374000	4331074	43200	169			17345087963136
14	1594323	96887416084	9655013170530	76788233223520	140278938424625	91666597450428	26928551389740	4053479921920	337607141157	16205174700	452888722	7184736	59319	196		349821975040000
15	4782969	678218289880	106423141562019	1168987966211520	2785281444479625	2310333412691304	851839311830507	160829863068160	16946435466675	1048800753000	39082476433	876657600	11457355	79576	225	7402408569806848

Technical details

Node id: n-26057d10e4649dfa · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e2e41084f3a7d696

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,        4;
   3:     9,       40,         9;
   4:    27,      316,       189,         16;
   5:    81,     2320,      2790,        576,        25;
   6:   243,    16564,     35910,      13600,      1375,        36;
   7:   729,   116920,    432279,     267840,     47375,      2808,       49;
   8:  2187,   821356,   5018139,    4786096,   1318625,    132804,     5145,     64;
   9:  6561,  5758240,  57047580,   80712576,  32532150,   4953312,   319676,   8704,    81;
  10: 19683, 40333924, 640479420, 1312106560, 744224250, 160772472, 15373260, 687360, 13851, 100;

%F Formula

The row polynomials satisfy P(n,x) = (3-12*x+9*x^2)*P(n-2,x) - (24*x-12*x^2-4*x^3)*d/dx P(n-2,x) - (16*x^2-16*x^3)*d^2/dx^2 P(n-2,x) + (2+4*x)*P(n-1,x) + (8*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[(3 - 12*x + 9*x^2)*P[n - 2, x] - (24*x - 12*x^2 - 4*x^3)*D[P[n - 2, x], x] - (16*x^2 - 16*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + (8*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 4*x; for(n = 3, N, my(i = n); v[i] = (3 - 12*x + 9*x^2)*v[i - 2] - (24*x - 12*x^2 - 4*x^3)*deriv(v[i - 2], x) - (16*x^2 - 16*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + (8*x - 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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.