A059110 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	7	6	1													14
4	34	39	12	1												86
5	209	292	126	20	1											648
6	1546	2505	1360	310	30	1										5752
7	13327	24306	15675	4580	645	42	1									58576
8	130922	263431	194796	68635	12530	1197	56	1								671568
9	1441729	3154824	2612092	1068536	237510	29624	2044	72	1							8546432
10	17572114	41368977	37705536	17453100	4541796	694134	62832	3276	90	1						119401856
11	234662231	589410910	583900245	300216120	89050710	16024932	1785210	122520	4995	110	1					1815177984
12	3405357682	9064804551	9664662700	5443344885	1805152140	372185814	49016352	4153050	223410	7315	132	1				29808908032
13	53334454417	149641946796	170364779706	103978789420	37991860335	8798594904	1332567852	133864632	8910495	385660	10362	156	1			525586164736
14	896324308634	2638693215769	3187414790352	2089971641186	831825249970	213160570623	36387004368	4215009084	333366462	17884295	636064	14274	182	1		9898343691264
15	16083557845279	49490245341642	63094659575311	44134205331580	18960454243731	5313358688638	1006792725939	131784135912	12043396365	768950182	33942909	1009372	19201	210	1	198227905206272

Technical details

Node id: n-d04cf1089b783b9c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ced16c5d17801fb2

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:        2,        1;
   3:        7,        6,        1;
   4:       34,       39,       12,        1;
   5:      209,      292,      126,       20,       1;
   6:     1546,     2505,     1360,      310,      30,      1;
   7:    13327,    24306,    15675,     4580,     645,     42,     1;
   8:   130922,   263431,   194796,    68635,   12530,   1197,    56,    1;
   9:  1441729,  3154824,  2612092,  1068536,  237510,  29624,  2044,   72,  1;
  10: 17572114, 41368977, 37705536, 17453100, 4541796, 694134, 62832, 3276, 90, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n + x)*v[i - 1] + (1 + x)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n an…?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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.