A225467 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 - 48\,x + 144\,x^{2}\right)\,P_{n-2}(x) + \left(-24\,x + 48\,x^{2} + 64\,x^{3}\right)\,P_{n-2}'(x) + \left(-16\,x^{2} + 64\,x^{3}\right)\,P_{n-2}''(x) + \left(2 + 16\,x\right)\,P_{n-1}(x) + \left(8\,x - 16\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 16\,x\)

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	16														19
3	9	160	144													313
4	27	1264	3024	1024												5339
5	81	9280	44640	36864	6400											97265
6	243	66256	574560	870400	352000	36864										1900323
7	729	467680	6916464	17141760	12128000	2875392	200704									39730729
8	2187	3285424	80290224	306310144	337568000	135991296	21073920	1048576								885569771
9	6561	23032960	912761280	5165604864	8328230400	5072191488	1309392896	142606336	5308416							20959135201
10	19683	161335696	10247670720	83974819840	190521408000	164631011328	62968872960	11261706240	907739136	26214400						524700798003
11	59049	1129664800	114176399184	1332494622720	4144749376000	4883916570624	2596483964928	678093127680	88783257600	5505024000	126877696					13845458944281
12	177147	7908598384	1266107374224	20799340401664	87078329536000	136203837530112	96695279493120	34586190938112	6540260474880	653131776000	32100057088	603979776				383863090336507
13	531441	55363023040	13998358501920	320993536241664	1784484138694400	3634259441319936	3352326772867072	1577355420106752	404001708244992	57769721856000	4541460250624	181193932800	2835349504			11149969950920145
14	1594323	387549664336	154480210728480	4914446926305280	35911408236704000	93866595789238272	110299346492375040	66412215040737280	22125421602865152	4248089316556800	474888244559872	30134966943744	995207675904	13153337344		338438422739285827
15	4782969	2712873159520	1702770264992304	74815229837537280	713032049786784000	2365781414595895296	3489133821257756672	2635036476508733440	1110601594744012800	274936824594432000	40980946808209408	3676968478310400	192222519623680	5340254961664	60397977600	10709898432927169033

Technical details

Node id: n-67bc02bc0a0fe6ee · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-089ec980a102281d

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,        16;
   3:     9,       160,         144;
   4:    27,      1264,        3024,        1024;
   5:    81,      9280,       44640,       36864,         6400;
   6:   243,     66256,      574560,      870400,       352000,        36864;
   7:   729,    467680,     6916464,    17141760,     12128000,      2875392,      200704;
   8:  2187,   3285424,    80290224,   306310144,    337568000,    135991296,    21073920,     1048576;
   9:  6561,  23032960,   912761280,  5165604864,   8328230400,   5072191488,  1309392896,   142606336,   5308416;
  10: 19683, 161335696, 10247670720, 83974819840, 190521408000, 164631011328, 62968872960, 11261706240, 907739136, 26214400;

%F Formula

The row polynomials satisfy P(n,x) = (3-48*x+144*x^2)*P(n-2,x) - (24*x-48*x^2-64*x^3)*d/dx P(n-2,x) - (16*x^2-64*x^3)*d^2/dx^2 P(n-2,x) + (2+16*x)*P(n-1,x) + (8*x-16*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 16*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[(3 - 48*x + 144*x^2)*P[n - 2, x] - (24*x - 48*x^2 - 64*x^3)*D[P[n - 2, x], x] - (16*x^2 - 64*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 16*x)*P[n - 1, x] + (8*x - 16*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 + 16*x; for(n = 3, N, my(i = n); v[i] = (3 - 48*x + 144*x^2)*v[i - 2] - (24*x - 48*x^2 - 64*x^3)*deriv(v[i - 2], x) - (16*x^2 - 64*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 16*x)*v[i - 1] + (8*x - 16*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, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …?
  - 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.