A257607 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + n\right)\,P_{n}(x) = \left(-100 - 120\,x - 60\,x^{2} - 20\,n - 64\,n x - 47\,n x^{2} - 8\,n^{2} x - 12\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-20\,x + 20\,x^{3} - 4\,n x - 5\,n x^{2} + 9\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(40 + 32\,x + 9\,n + 16\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5 + 10\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	10														15
2	25	120	75													220
3	125	1070	1605	500												3300
4	625	8420	22470	16840	3125											51480
5	3125	61770	258330	344440	154425	18750										840840
6	15625	433120	2642205	5511040	4403675	1299360	109375									14414400
7	78125	2942470	24992235	75794780	94743475	49984470	10298645	625000								259459200
8	390625	19529820	223496400	939588040	1705382550	1409382060	521491600	78119280	3515625							4900896000
9	1953125	127335170	1916011560	10794651520	27093293450	32511952140	18890640160	5109364160	573008265	19531250						97037740800
10	9765625	818698520	15895116735	117013397120	392266099450	650239042800	549172539230	234026794240	47685350205	4093492600	107421875					2011327718400
11	48828125	5205159870	128458486065	1211622533700	5285595851850	11680302940740	13627020097530	8456953362960	2726150700825	428194953550	28628379285	585937500				43578767232000
12	244140625	32793459220	1016325499530	12090872009480	67259228839275	192915609674040	299794442145660	257220812898720	121066611910695	30227180023700	3726526831610	196760755320	3173828125			985550582016000
13	1220703125	205061536570	7901321517990	117053486066440	816923319719375	2978400066633090	5998557399038820	6855494170330080	4467600099949635	1633846639438750	321897086682710	31605286071960	1332899987705	17089843750		23230835147520000
14	6103515625	1274314531920	60538319808465	1104989414248640	9547062741220125	43508312437616400	111156532400028885	164531860087921920	142915541657179995	72513854062694000	21003538030684275	3314968242745920	262332719170015	8920201723440	91552734375	569929822285824000

Technical details

Node id: n-994b7fb19b3cbd3e · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-df7467a067c1cbae

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:       5,        10;
  2:      25,       120,         75;
  3:     125,      1070,       1605,         500;
  4:     625,      8420,      22470,       16840,        3125;
  5:    3125,     61770,     258330,      344440,      154425,       18750;
  6:   15625,    433120,    2642205,     5511040,     4403675,     1299360,      109375;
  7:   78125,   2942470,   24992235,    75794780,    94743475,    49984470,    10298645,     625000;
  8:  390625,  19529820,  223496400,   939588040,  1705382550,  1409382060,   521491600,   78119280,   3515625;
  9: 1953125, 127335170, 1916011560, 10794651520, 27093293450, 32511952140, 18890640160, 5109364160, 573008265, 19531250;

%F Formula

The row polynomials satisfy (n+4)*P(n,x) = -(20*n+100+(8*n^2+64*n+120)*x+(n^3+12*n^2+47*n+60)*x^2)*P(n-2,x) - ((4*n+20)*x+(n^2+5*n)*x^2-(n^2+9*n+20)*x^3)*d/dx P(n-2,x) + (9*n+40+(2*n^2+16*n+32)*x)*P(n-1,x) + ((n+4)*x-(n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 100 + (8*n^2 + 64*n + 120)*x + (n^3 + 12*n^2 + 47*n + 60)*x^2)*P[n - 2, x] - ((4*n + 20)*x + (n^2 + 5*n)*x^2 - (n^2 + 9*n + 20)*x^3)*D[P[n - 2, x], x] + (9*n + 40 + (2*n^2 + 16*n + 32)*x)*P[n - 1, x] + ((n + 4)*x - (n + 4)*x^2)*D[P[n - 1, x], x])/(n + 4)]];
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] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 100 + (8*n^2 + 64*n + 120)*x + (n^3 + 12*n^2 + 47*n + 60)*x^2)*v[i - 2] - ((4*n + 20)*x + (n^2 + 5*n)*x^2 - (n^2 + 9*n + 20)*x^3)*deriv(v[i - 2], x) + (9*n + 40 + (2*n^2 + 16*n + 32)*x)*v[i - 1] + ((n + 4)*x - (n + 4)*x^2)*deriv(v[i - 1], x))/(n + 4)); 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 + 5.?
  - 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 + 5.?
    
    ↑ 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.