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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	6531840	32901120	54190080	40642560	15805440	3354624	387072	22528	512							153835776
9	72576000	413683200	783820800	690923520	325140480	87091200	13547520	1198080	55296	1024						2388037120
10	878169600	5588352000	11975040000	12134707200	6706022400	2171473920	425779200	50688000	3548160	133120	2048					39933915648
11	11496038400	80791603200	193197312000	221298739200	140507136000	53392711680	12645642240	1895731200	178421760	10137600	315392	4096				715413792768
12	161902540800	1245404160000	3287866982400	4201163366400	3013878067200	1315146792960	365318553600	66105262080	7828254720	597442560	28114944	737280	8192			13665240283136
13	2440992153600	20399720140800	58932524851200	83109970944000	66487976755200	32769074257920	10448110632960	2216265891840	316609413120	30381711360	1918844928	76038144	1703936	16384		277153623355392
14	39230231040000	353943862272000	1110651429888000	1713576491827200	1512601471180800	831099709440000	299195895398400	72820165017600	12189462405120	1410351022080	111932620800	5948866560	201277440	3899392	32768	5946837156188160

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.