A257616 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(8 - 80\,x - 40\,x^{2} + 24\,n - 192\,n x - 36\,n x^{2} + 144\,n^{2} x + 216\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(24\,x - 24\,x^{3} + 72\,n x - 36\,n x^{2} - 36\,n x^{3} - 108\,n^{2} x^{2} + 108\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 + 16\,x - 6\,n - 48\,n x + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 12\,x^{2} + 18\,n x - 18\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	64	12													80
3	8	624	936	32												1600
4	16	5312	26208	10624	80											42240
5	32	43328	526272	701696	108320	192										1379840
6	64	348672	9057600	28067840	15096000	1046016	448									53616640
7	128	2794240	143542656	875353600	1094192000	287085312	9779840	1024								2412748800
8	256	22365184	2168868864	23631558656	56897984000	35447337984	5060694016	89460736	2304							123318272000
9	512	178947072	31840266240	582520528896	2424609930240	2909531916288	1019410925568	84907376640	805261824	5120						7053805158400
10	1024	1431633920	459184757760	13518372864000	90709583308800	186210042642432	126993416632320	27036745728000	1377554273280	7158169600	11264					446313490022400
11	2048	11453198336	6548843857920	300979774464000	3101959673548800	10095078363439104	11777591424012288	4963135477678080	677204492544000	21829479526400	62992590848	24576				30944401974886400
12	4096	91625865216	92748961456128	6508575830671360	99462187416268800	486825978393427968	895096948224933888	649101304524570624	179031937349283840	16271439576678400	340079525339136	549755191296	53248			2332731841183744000
13	8192	733007527936	1307831298637824	137838764093800448	3041617180969984000	21546162553565822976	59004084256743456768	67433239150563950592	32319243830348734464	6083234361939968000	379056601257951232	5231325194551296	4764548931584	114688		189951021353533440000
14	16384	5864061534208	18391002016530432	2875351986285838336	89764550922489118720	893873876275289260032	3499014684047591030784	5934125045249627652096	4498733165204045611008	1489789793792148766720	197482012029476061184	8626055958857515008	79694342071631872	41048430739456	245760	16614382667722391552000

Technical details

Node id: n-ad646368add3e231 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-506bdfb5cd7829bb

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:   2,         4;
  2:   4,        64,          12;
  3:   8,       624,         936,           32;
  4:  16,      5312,       26208,        10624,            80;
  5:  32,     43328,      526272,       701696,        108320,           192;
  6:  64,    348672,     9057600,     28067840,      15096000,       1046016,           448;
  7: 128,   2794240,   143542656,    875353600,    1094192000,     287085312,       9779840,        1024;
  8: 256,  22365184,  2168868864,  23631558656,   56897984000,   35447337984,    5060694016,    89460736,      2304;
  9: 512, 178947072, 31840266240, 582520528896, 2424609930240, 2909531916288, 1019410925568, 84907376640, 805261824, 5120;

%F Formula

The row polynomials satisfy (3*n-2)*P(n,x) = (24*n+8+(144*n^2-192*n-80)*x-(108*n^3-216*n^2+36*n+40)*x^2)*P(n-2,x) + ((72*n+24)*x-(108*n^2+36*n)*x^2+(108*n^2-36*n-24)*x^3)*d/dx P(n-2,x) - (6*n+8-(36*n^2-48*n+16)*x)*P(n-1,x) + ((18*n-12)*x-(18*n-12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 8 + (144*n^2 - 192*n - 80)*x - (108*n^3 - 216*n^2 + 36*n + 40)*x^2)*P[n - 2, x] + ((72*n + 24)*x - (108*n^2 + 36*n)*x^2 + (108*n^2 - 36*n - 24)*x^3)*D[P[n - 2, x], x] - (6*n + 8 - (36*n^2 - 48*n + 16)*x)*P[n - 1, x] + ((18*n - 12)*x - (18*n - 12)*x^2)*D[P[n - 1, x], x])/(3*n - 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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((24*n + 8 + (144*n^2 - 192*n - 80)*x - (108*n^3 - 216*n^2 + 36*n + 40)*x^2)*v[i - 2] + ((72*n + 24)*x - (108*n^2 + 36*n)*x^2 + (108*n^2 - 36*n - 24)*x^3)*deriv(v[i - 2], x) - (6*n + 8 - (36*n^2 - 48*n + 16)*x)*v[i - 1] + ((18*n - 12)*x - (18*n - 12)*x^2)*deriv(v[i - 1], x))/(3*n - 2)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142458 → θ+1

References

[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.