A299989 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 9\,x + x^{2} - x^{3} + 15\,n x + 17\,n x^{2} + 5\,n x^{3} + 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n}(x) = \left(27 + 108\,x + 171\,x^{2} + 136\,x^{3} + 57\,x^{4} + 12\,x^{5} + x^{6} + 45\,n x + 135\,n x^{2} + 154\,n x^{3} + 83\,n x^{4} + 21\,n x^{5} + 2\,n x^{6} + 12\,n^{2} x^{2} + 28\,n^{2} x^{3} + 23\,n^{2} x^{4} + 8\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 4\,x\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	x²⁹	P_n(1)
0	4																													4
1	12	36	16																											64
2	36	216	352	200	36																									840
3	108	972	2736	3400	2052	588	64																							9920
4	324	3888	15552	30000	31896	19600	6912	1296	100																					109568
5	972	14580	75600	198000	303480	291256	179840	71280	17500	2420	144																			1155072
6	2916	52488	334368	1107000	2221020	2914128	2600704	1605744	685500	198440	37152	4056	196																	11763712
7	8748	183708	1388016	5556600	13791708	23004324	26975424	22800528	14049700	6311844	2043216	463736	69972	6300	256															116654080
8	26244	629856	5505408	25855200	76504176	154992096	225630720	242992224	196893400	120996128	56407680	19798688	5142256	957600	120832	9248	324													1132462080
9	78732	2125764	21100176	113724000	390825648	932175216	1622654208	2127443616	2143189800	1678940824	1028731104	493193376	184051056	52844400	11437056	1803360	195372	12996	400											10804527104
10	236196	7085880	78732000	478953000	1875002580	5147253216	10432903680	16137960480	19470861000	18587894160	14164699968	8653868080	4240320840	1660284000	515205120	124931232	23148180	3162360	300000	17640	484									101602820096
11	708588	23383404	287529264	1948617000	8560924020	26588854908	61523820864	109739181024	154407627000	174109973400	159016660704	118418605552	72146447912	35973135000	14640130560	4833809376	1281746268	268709628	43494000	5239080	441892	23276	576							943819063296
12	2125764	76527504	1031704128	7707862800	37553746824	130195329264	338704084224	684634891248	1101275732700	1433594204064	1528217995392	1344929088672	982228904944	596862021600	301870221312	126816261088	44051029308	12556683216	2903841600	535783248	76953096	8286256	628992	30000	676					8675833937920
13	6377292	248714388	3647811024	29784315600	159379864488	610152687144	1763078638464	3983814950544	7213965372900	10653683354028	12994873368048	13211214475680	11265958874352	8091406897200	4904907265024	2510217609120	1082906114004	392406865708	118748401200	29752618896	6096608232	1003986984	129557376	12610000	870012	37908	784			79027398246400
14	19131876	803538792	12737577888	112842639000	657904826124	2756570137776	8760314384640	21885650206992	44069866740900	72827773442424	100087784885088	115517292423624	112772870660268	93594559514400	66254335424512	40072337332832	20713384406988	9139106413528	3432365736800	1092305141928	292590061764	65355376944	12016892160	1786330000	209215916	18574920	1174432	47096	900	714132802240512

Technical details

Node id: n-3b0976fc57c747e3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-383c3e2053bee1bd

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 0,     4;
  1: 0,    12,      36,       16;
  2: 0,    36,     216,      352,       200,        36;
  3: 0,   108,     972,     2736,      3400,      2052,       588,         64;
  4: 0,   324,    3888,    15552,     30000,     31896,     19600,       6912,       1296,        100;
  5: 0,   972,   14580,    75600,    198000,    303480,    291256,     179840,      71280,      17500,       2420,        144;
  6: 0,  2916,   52488,   334368,   1107000,   2221020,   2914128,    2600704,    1605744,     685500,     198440,      37152,      4056,       196;
  7: 0,  8748,  183708,  1388016,   5556600,  13791708,  23004324,   26975424,   22800528,   14049700,    6311844,    2043216,    463736,     69972,     6300,      256;
  8: 0, 26244,  629856,  5505408,  25855200,  76504176, 154992096,  225630720,  242992224,  196893400,  120996128,   56407680,  19798688,   5142256,   957600,   120832,    9248,    324;
  9: 0, 78732, 2125764, 21100176, 113724000, 390825648, 932175216, 1622654208, 2127443616, 2143189800, 1678940824, 1028731104, 493193376, 184051056, 52844400, 11437056, 1803360, 195372, 12996, 400;

%F Formula

The row polynomials satisfy (9+(15*n+9)*x+(4*n^2+17*n+1)*x^2+(4*n^2+5*n-1)*x^3+n^2*x^4)*P(n,x) = (27+(45*n+108)*x+(12*n^2+135*n+171)*x^2+(28*n^2+154*n+136)*x^3+(23*n^2+83*n+57)*x^4+(8*n^2+21*n+12)*x^5+(n^2+2*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 4*x.

%t Mathematica

P[0, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((27 + (45*n + 108)*x + (12*n^2 + 135*n + 171)*x^2 + (28*n^2 + 154*n + 136)*x^3 + (23*n^2 + 83*n + 57)*x^4 + (8*n^2 + 21*n + 12)*x^5 + (n^2 + 2*n + 1)*x^6)*P[n - 1, x])/(9 + (15*n + 9)*x + (4*n^2 + 17*n + 1)*x^2 + (4*n^2 + 5*n - 1)*x^3 + n^2*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (45*n + 108)*x + (12*n^2 + 135*n + 171)*x^2 + (28*n^2 + 154*n + 136)*x^3 + (23*n^2 + 83*n + 57)*x^4 + (8*n^2 + 21*n + 12)*x^5 + (n^2 + 2*n + 1)*x^6)*v[i - 1])/(9 + (15*n + 9)*x + (4*n^2 + 17*n + 1)*x^2 + (4*n^2 + 5*n - 1)*x^3 + n^2*x^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

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,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…?
  - Parent-row provers of polytool polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.