A397048 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	9	7	1													17
3	64	61	15	1												141
4	625	671	217	26	1											1540
5	7776	9031	3465	565	40	1										20878
6	117649	144495	62401	12410	1220	57	1									338233
7	2097152	2685817	1265607	289821	35350	2324	77	1								6376149
8	43046721	56953279	28702801	7309722	1041831	85974	4046	100	1							137144475
9	1000000000	1357947691	721942485	199833565	32080860	3125451	186186	6582	126	1						3315122947
10	25937424601	35979939623	19980594001	5915641250	1043778730	115589187	8190987	368880	10155	155	1					88981537570
11	743008370688	1049152023349	604126376139	189088860301	36038350810	4422221518	358651755	19325427	681120	15015	187	1				2626214876310
12	23298085122481	33395827252815	19828299131257	6503269173698	1321783056341	176483835264	15958797184	989417484	41915742	1187725	21439	222	1			84540738911653
13	793714773254144	1152480295105231	702462131952945	239774930436405	51466041586960	7376697981653	729931009808	50704331392	2482059294	84853912	1975259	29731	260	1		2948058074576995
14	29192926025390625	42864668012537311	26727782254467841	9443290665378570	2124280195193520	323475212305245	34538371961093	2637199055920	145212929290	5756300550	162164002	3156426	40222	301	1	110713749070880917

Technical details

Node id: n-feabd47ffc664156 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0c9a431d23aaf6f8

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:          2,          1;
  2:          9,          7,         1;
  3:         64,         61,        15,         1;
  4:        625,        671,       217,        26,        1;
  5:       7776,       9031,      3465,       565,       40,       1;
  6:     117649,     144495,     62401,     12410,     1220,      57,      1;
  7:    2097152,    2685817,   1265607,    289821,    35350,    2324,     77,    1;
  8:   43046721,   56953279,  28702801,   7309722,  1041831,   85974,   4046,  100,   1;
  9: 1000000000, 1357947691, 721942485, 199833565, 32080860, 3125451, 186186, 6582, 126, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 1 + x)*P[n - 1, x] + (n + 1 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 1 + x)*v[i - 1] + (n + 1 + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), 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 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 Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.?
  - 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.