A132166 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(-3276 - 252\,x - 6\,x^{2} - 720\,n - 36\,n x - 36\,n^{2}\right)\,P_{n-2}(x) + \left(-1512\,x - 114\,x^{2} - 2\,x^{3} - 216\,n x - 6\,n x^{2}\right)\,P_{n-2}'(x) + \left(252 + 12\,x + 150\,n + 6\,n x + 12\,n^{2}\right)\,P_{n-1}(x) + \left(72\,x + 2\,x^{2} + 36\,n x + n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 42 + 2\,x\)

\(P_{1}(x) = 1113 + 126\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	42	2															44
1	1113	126	3														1242
2	23184	4662	252	4													28102
3	412272	130410	11970	420	5												555077
4	6531840	3036096	419580	24360	630	6											10012512
5	94618368	61825680	12032307	1025640	43155	882	7										169546039
6	1274921856	1134724032	297906336	34895700	2120580	69678	1176	8									2744639366
7	16176101760	19147482432	6579379872	1015163604	83427435	3913434	105252	1512	9								43005575310
8	195030291456	301257605376	132461671680	26136527040	2785427190	174707442	6650280	151200	1890	10							657853033564
9	2250184914432	4465750423680	2468708057088	609388621920	81792868365	6579546246	332066826	10614240	208845	2310	11						9882747323963
10	24982689005568	62872014288384	43079649451776	13078768187520	2163596981040	216833313690	13919473512	586060272	16125480	279510	2772	12					146408073169536
11	268115731513344	846006485119488	710065304573184	261537132528000	52442019559440	6409229154174	508450349988	27054854064	975438450	23541210	364518	3276	13				2145112406999149
12	2791778491072512	10936129302134784	11131395223560192	4918638112303104	1179608957452800	172934264006784	16597533486744	1084543699104	49164118668	1548120420	33255684	465192	3822	14			31148217173679824
13	28292068847112192	136381517048070144	166897703632868352	87640099488264960	24865632926426880	4315764595807728	492995049294951	38732411170080	2145310625970	84580081740	2362166037	45700200	582855	4410	15		448926746298176514
14	279779813592072192	1646563640640528384	2404337527892754432	1488312871547857920	495022248188657280	100632787995475008	13505327558243136	1255201112514888	83176545359160	3990227260140	139040043912	3486748356	61343100	718830	5040	16	6429496727889581794

Technical details

Node id: n-457c0b39dd0a8002 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-db5749df03dec5cf

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, 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:            42,             2;
  1:          1113,           126,             3;
  2:         23184,          4662,           252,            4;
  3:        412272,        130410,         11970,          420,           5;
  4:       6531840,       3036096,        419580,        24360,         630,          6;
  5:      94618368,      61825680,      12032307,      1025640,       43155,        882,         7;
  6:    1274921856,    1134724032,     297906336,     34895700,     2120580,      69678,      1176,        8;
  7:   16176101760,   19147482432,    6579379872,   1015163604,    83427435,    3913434,    105252,     1512,      9;
  8:  195030291456,  301257605376,  132461671680,  26136527040,  2785427190,  174707442,   6650280,   151200,   1890,   10;
  9: 2250184914432, 4465750423680, 2468708057088, 609388621920, 81792868365, 6579546246, 332066826, 10614240, 208845, 2310, 11;

%F Formula

The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+720*n+3276+(36*n+252)*x+6*x^2)*P(n-2,x) - ((216*n+1512)*x+(6*n+114)*x^2+2*x^3)*d/dx P(n-2,x) + (12*n^2+150*n+252+(6*n+12)*x)*P(n-1,x) + ((36*n+72)*x+(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 42 + 2*x, P(1,x) = 1113 + 126*x + 3*x^2.

%t Mathematica

P[0, x_] := 42 + 2*x;
P[1, x_] := 1113 + 126*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 720*n + 3276 + (36*n + 252)*x + 6*x^2)*P[n - 2, x] - ((216*n + 1512)*x + (6*n + 114)*x^2 + 2*x^3)*D[P[n - 2, x], x] + (12*n^2 + 150*n + 252 + (6*n + 12)*x)*P[n - 1, x] + ((36*n + 72)*x + (n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 42 + 2*x; v[2] = 1113 + 126*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 720*n + 3276 + (36*n + 252)*x + 6*x^2)*v[i - 2] - ((216*n + 1512)*x + (6*n + 114)*x^2 + 2*x^3)*deriv(v[i - 2], x) + (12*n^2 + 150*n + 252 + (6*n + 12)*x)*v[i - 1] + ((36*n + 72)*x + (n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n + 6)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from A convolution triangle of numbers obtained from A036224.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.