A086915 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	24	16													40
3	144	192	48												384
4	960	1920	960	128											3968
5	7200	19200	14400	3840	320										44960
6	60480	201600	201600	80640	13440	768									558528
7	564480	2257920	2822400	1505280	376320	43008	1792								7571200
8	5806080	27095040	40642560	27095040	9031680	1548288	129024	4096							111351808
9	65318400	348364800	609638400	487710720	203212800	46448640	5806080	368640	9216						1766877696
10	798336000	4790016000	9580032000	8941363200	4470681600	1277337600	212889600	20275200	1013760	20480					30091965440
11	10538035200	70253568000	158070528000	168608563200	98354995200	33721712640	7025356800	892108800	66908160	2703360	45056				547534524416
12	149448499200	1095955660800	2739889152000	3287866982400	2191911321600	876764528640	219191132160	34792243200	3479224320	210862080	7028736	98304			10599516733440
13	2266635571200	18133084569600	49865982566400	66487976755200	49865982566400	22795877744640	6648797675520	1266437652480	158304706560	12792299520	639614976	17891328	212992		217502529826816
14	36614882304000	317328979968000	951986939904000	1396247511859200	1163539593216000	598391790796800	199463930265600	44325317836800	6648797675520	671595724800	44773048320	1878589440	44728320	458752	4715266036375552

Technical details

Node id: n-98fbd6b8b374e38e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-18b5860a7998f46b

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:        4;
  2:       24,        16;
  3:      144,       192,        48;
  4:      960,      1920,       960,       128;
  5:     7200,     19200,     14400,      3840,       320;
  6:    60480,    201600,    201600,     80640,     13440,      768;
  7:   564480,   2257920,   2822400,   1505280,    376320,    43008,    1792;
  8:  5806080,  27095040,  40642560,  27095040,   9031680,  1548288,  129024,   4096;
  9: 65318400, 348364800, 609638400, 487710720, 203212800, 46448640, 5806080, 368640, 9216;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n + 2*n*x)*v[i - 1] + n*x*deriv(v[i - 1], x))/(n - 1)); 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.