A257606 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-48 - 48\,x - 24\,x^{2} - 12\,n - 36\,n x - 26\,n x^{2} - 6\,n^{2} x - 9\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 12\,x^{3} - 3\,n x - 4\,n x^{2} + 7\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(24 + 18\,x + 7\,n + 12\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	80	48													144
3	64	592	888	256												1800
4	256	3856	10656	7712	1280											23760
5	1024	23376	104424	139232	58440	6144										332640
6	4096	135312	907056	1949248	1511760	405936	28672									4942080
7	16384	758480	7269048	23320000	29150000	14538096	2654680	131072								77837760
8	65536	4152848	54991488	250468576	466400000	375702864	128313472	16611392	589824							1297296000
9	262144	22337104	398470920	2486499872	6548971480	7858765776	4351374776	1062589120	100516968	2621440						22832409600
10	1048576	118501264	2792893392	23249739264	83473020240	141457783968	116862228336	46499478528	8378680176	592506320	11534336					423437414400
11	4194304	621866448	19068135000	207434469120	987468076800	2274796298592	2653929015024	1579948922880	466727555520	63560450000	3420265464	50331648				8257029580800
12	16777216	3235161360	127468005408	1782555623840	11011261651200	33507745301088	53078580300480	44676993734784	19820270972160	4456389059600	467382686496	19410968160	218103808			168951528345600
13	67108864	16712677712	837599163048	14857292134496	117056622097000	460131875487072	960801867702096	1098059277373824	690197813230608	234113244194000	40857553369864	3350396652192	108632405128	939524096		3620389893120000
14	268435456	85845089936	5426699243376	120753028202432	1196455589129680	5967270184096848	16049864933839968	24157304102224128	20635540629222816	9945450306828080	2632202296085296	362259084607296	23515696721296	600915629552	4026531840	81096733605888000

Technical details

Node id: n-b15ed6e1650924b8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3dd8844cb1c111f1

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,       80,        48;
  3:     64,      592,       888,        256;
  4:    256,     3856,     10656,       7712,       1280;
  5:   1024,    23376,    104424,     139232,      58440,       6144;
  6:   4096,   135312,    907056,    1949248,    1511760,     405936,      28672;
  7:  16384,   758480,   7269048,   23320000,   29150000,   14538096,    2654680,     131072;
  8:  65536,  4152848,  54991488,  250468576,  466400000,  375702864,  128313472,   16611392,    589824;
  9: 262144, 22337104, 398470920, 2486499872, 6548971480, 7858765776, 4351374776, 1062589120, 100516968, 2621440;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = -(12*n+48+(6*n^2+36*n+48)*x+(n^3+9*n^2+26*n+24)*x^2)*P(n-2,x) - ((3*n+12)*x+(n^2+4*n)*x^2-(n^2+7*n+12)*x^3)*d/dx P(n-2,x) + (7*n+24+(2*n^2+12*n+18)*x)*P(n-1,x) + ((n+3)*x-(n+3)*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[(-(12*n + 48 + (6*n^2 + 36*n + 48)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*P[n - 2, x] - ((3*n + 12)*x + (n^2 + 4*n)*x^2 - (n^2 + 7*n + 12)*x^3)*D[P[n - 2, x], x] + (7*n + 24 + (2*n^2 + 12*n + 18)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(n + 3)]];
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] = (-(12*n + 48 + (6*n^2 + 36*n + 48)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*v[i - 2] - ((3*n + 12)*x + (n^2 + 4*n)*x^2 - (n^2 + 7*n + 12)*x^3)*deriv(v[i - 2], x) + (7*n + 24 + (2*n^2 + 12*n + 18)*x)*v[i - 1] + ((n + 3)*x - (n + 3)*x^2)*deriv(v[i - 1], x))/(n + 3)); 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) = 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) = x + 4.?
    
    ↑ Liu–Wang theorem (above)

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.