A112486 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	2	10	9													21
3	6	52	105	60												223
4	24	308	1020	1260	525											3137
5	120	2088	9912	19600	17325	5670										54715
6	720	16056	101220	282128	393750	270270	72765									1136909
7	5040	138528	1103652	4035920	7857850	8440740	4729725	1081080								27392535
8	40320	1327392	12906648	59110480	149788100	222882660	194234040	91891800	18243225							750424665
9	362880	13999680	161884008	898623264	2833902500	5457051600	6553927380	4799995200	1964187225	344594250						23028527987
10	3628800	161254080	2173602816	14272246560	54277263040	129465336000	200361241080	201443161920	127155278250	45831035250	7202019825					782346067621
11	39916800	2014882560	31159410048	237487233984	1063923260400	3048436110720	5822827370360	7518709598400	6495174315630	3601009912500	1159525191825	164991726900				29145298930127
12	479001600	27151476480	475521950592	4143950830656	21475657803600	72250740481920	165271542195760	262639359702720	290615903978400	219958415176500	108692883198900	31623414322500	4111043861925			1181286063981553
13	6227020800	392574712320	7703010264960	75813652302336	447965199274320	1738616747868000	4657424851839760	8841540681733760	12012319576857600	11620122893379000	7822929960846000	3486165194912400	924984868933125	110681950128750		51746667390073131
14	87178291200	6062977267200	132081095266560	1453162974792192	9674413698362400	42709766513817312	131687834947869200	292127779501317760	473152429296827280	560897305093125000	481739428262792700	291972659756778000	118466580621138750	28887988983603750	3201870700153125	2436109451601402429

Technical details

Node id: n-f352cbd6d9928aaa · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9df8a03f2559bb60

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:      1,        2;
  2:      2,       10,         9;
  3:      6,       52,       105,        60;
  4:     24,      308,      1020,      1260,        525;
  5:    120,     2088,      9912,     19600,      17325,       5670;
  6:    720,    16056,    101220,    282128,     393750,     270270,      72765;
  7:   5040,   138528,   1103652,   4035920,    7857850,    8440740,    4729725,    1081080;
  8:  40320,  1327392,  12906648,  59110480,  149788100,  222882660,  194234040,   91891800,   18243225;
  9: 362880, 13999680, 161884008, 898623264, 2833902500, 5457051600, 6553927380, 4799995200, 1964187225, 344594250;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x + (n^2 - 2*n + 1)*x^2)*P[n - 2, x] - ((n - 1)*x + (2*n - 2)*x^2 + (n - 1)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 2*n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x + (n^2 - 2*n + 1)*x^2)*v[i - 2] - ((n - 1)*x + (2*n - 2)*x^2 + (n - 1)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 2*n*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); 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 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? 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
[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.