A086915 → D+1

nonnegative coefficients

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

Recurrence

\(\left(x + 2\,n\right)\,P_{n}(x) = \left(2 + 3\,x + 2\,x^{2} + 4\,n + 5\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(x + x^{2} + 2\,n x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	8	4							12
2	36	40	8						84
3	192	336	144	16					688
4	1200	2880	1920	448	32				6480
5	8640	26400	24000	8640	1280	64			69024
6	70560	262080	302400	147840	33600	3456	128		820064
7	645120	2822400	3951360	2446080	752640	118272	8960	256	10745088

All 15 rows

Technical details

Node id: n-32c953bc446d5813 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e58e8e6744652fd9

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:        2;
  1:        8,         4;
  2:       36,        40,         8;
  3:      192,       336,       144,        16;
  4:     1200,      2880,      1920,       448,        32;
  5:     8640,     26400,     24000,      8640,      1280,       64;
  6:    70560,    262080,    302400,    147840,     33600,     3456,      128;
  7:   645120,   2822400,   3951360,   2446080,    752640,   118272,     8960,     256;
  8:  6531840,  32901120,  54190080,  40642560,  15805440,  3354624,   387072,   22528,   512;
  9: 72576000, 413683200, 783820800, 690923520, 325140480, 87091200, 13547520, 1198080, 55296, 1024;

%F Formula

The row polynomials satisfy (2*n+x)*P(n,x) = (2*n^2+4*n+2+(5*n+3)*x+2*x^2)*P(n-1,x) + ((2*n+1)*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 4*n + 2 + (5*n + 3)*x + 2*x^2)*P[n - 1, x] + ((2*n + 1)*x + x^2)*D[P[n - 1, x], x])/(2*n + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 4*n + 2 + (5*n + 3)*x + 2*x^2)*v[i - 1] + ((2*n + 1)*x + x^2)*deriv(v[i - 1], x))/(2*n + 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 read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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.