A257613 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-16 - 8\,n - 16\,n x - 8\,n x^{2} - 8\,n^{2} x - 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x + 8\,x^{3} - 4\,n x - 8\,n x^{2} + 12\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(8 + 4\,x + 6\,n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	8														12
2	16	96	48													160
3	64	832	1248	256												2400
4	256	6272	19968	12544	1280											40320
5	1024	43776	253824	338432	109440	6144										752640
6	4096	291328	2818560	6768640	4697600	873984	28672									15482880
7	16384	1879040	28666368	112783360	140979200	57332736	6576640	131072								348364800
8	65536	11864064	274427904	1662939136	3383500800	2494408704	640331776	47456256	589824							8515584000
9	262144	73805824	2515752960	22483853312	69703444480	83644133376	39346743296	6708674560	332126208	2621440						224811417600
10	1048576	454369280	22340198400	285216604160	1286118400000	2342035734528	1800565760000	570433208320	67020595200	2271846400	11534336					6376469299200
11	4194304	2776547328	193715773440	3447904665600	21850794393600	57481973563392	67062302490624	34961271029760	7757785497600	645719244800	15271010304	50331648				193419568742400
12	16777216	16877387776	1649681891328	40161376010240	348407149363200	1276724788789248	2145993679699968	1702299718385664	627132868853760	100403440025600	6048833601536	101264326656	218103808			6248939913216000
13	67108864	102203850752	13855673253888	454403580624896	5285323632640000	26235918627766272	61147119439773696	69882422216884224	39353877941649408	10570647265280000	1249609846718464	55422693015552	664325029888	939524096		214249368453120000
14	268435456	617249636352	115137947762688	5024365812383744	77055991010426880	506835404690423808	1590525345684258816	2515767199807832064	2044961158736904192	844725674484039680	169523180222939136	15073097437151232	498931106971648	4320747454464	4026531840	7770110429233152000

Technical details

Node id: n-db27be2f5676b05b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7c3b635378939853

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:      4,        8;
  2:     16,       96,         48;
  3:     64,      832,       1248,         256;
  4:    256,     6272,      19968,       12544,        1280;
  5:   1024,    43776,     253824,      338432,      109440,        6144;
  6:   4096,   291328,    2818560,     6768640,     4697600,      873984,       28672;
  7:  16384,  1879040,   28666368,   112783360,   140979200,    57332736,     6576640,     131072;
  8:  65536, 11864064,  274427904,  1662939136,  3383500800,  2494408704,   640331776,   47456256,    589824;
  9: 262144, 73805824, 2515752960, 22483853312, 69703444480, 83644133376, 39346743296, 6708674560, 332126208, 2621440;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -(8*n+16+(8*n^2+16*n)*x+(4*n^3+12*n^2+8*n)*x^2)*P(n-2,x) - ((4*n+8)*x+(4*n^2+8*n)*x^2-(4*n^2+12*n+8)*x^3)*d/dx P(n-2,x) + (6*n+8+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 16 + (8*n^2 + 16*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*P[n - 2, x] - ((4*n + 8)*x + (4*n^2 + 8*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*D[P[n - 2, x], x] + (6*n + 8 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n + 16 + (8*n^2 + 16*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*v[i - 2] - ((4*n + 8)*x + (4*n^2 + 8*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 2], x) + (6*n + 8 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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) = 2*x + 4.?
  - 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) = 2*x + 4.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.