A204420 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 24\,x - 3\,x^{2} + 176\,n - 76\,n x + 8\,n x^{2} - 204\,n^{2} + 64\,n^{2} x - 4\,n^{2} x^{2} + 96\,n^{3} - 16\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(6 - 2\,x - 16\,n + 4\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	6	6													12
3	120	180	45												345
4	5040	9240	3780	420											18480
5	362880	756000	396900	75600	4725										1596105
6	39916800	91143360	56133000	14137200	1559250	62370									202951980
7	6227020800	15256200960	10534043520	3178375200	472972500	34054020	945945								35703612945
8	1307674368000	3390612825600	2555413660800	878139662400	158918760000	15664849200	794593800	16216200							8307234936000
9	355687428096000	966707617075200	781533142790400	296776697817600	61886370546000	7502781686400	526815689400	19848628800	310134825						2470641012464625
10	121645100408832000	344130127307366400	294835501920960000	121296470270976000	28213585299900000	3976960585197600	346482626490000	18227657448000	530330550750	6547290750					914462992955011500
11	51090942171709440000	149643747686264832000	134671009816985241600	59199911567596800000	15033738170571120000	2381305795340472000	242958237463141200	15971199163920000	653367238524000	15124241632500	151242416325				412280253287817539625
12	25852016738884976640000	78069919669148639232000	73306240262570668953600	34085066220924861542400	9309068971877394720000	1619871905950041744000	186835240331318779200	14467757590260374400	743858601059574000	24350029028325000	459171975962700	3794809718700			222444255438875035566000
13	15511210043330985984000000	48134552638433432371200000	46911366145135475343360000	22894581074640605890560000	6650599702530338755200000	1251195212726346888000000	159347408122334151720000	14018804277908189760000	853220842861817430000	35272756335316500000	945128983856557500	14799757902930000	102776096548125		141527760482828142563765625
14	10888869450418352160768000000	34628061294520142767718400000	34881228415741657702072320000	17760805095622582447534080000	5441413102445418254956800000	1093818469695878491044480000	151274529702758501479440000	14758206339437860178880000	1024782211630457099820000	50358100233246705900000	1711081409826203415000	38227774663268190000	505041738437486250	2988412653476250	104861305454615306530120687500

Technical details

Node id: n-df9ff482d15e6222 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-959faa625dfaa291

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:               1;
  2:               6,               6;
  3:             120,             180,              45;
  4:            5040,            9240,            3780,             420;
  5:          362880,          756000,          396900,           75600,           4725;
  6:        39916800,        91143360,        56133000,        14137200,        1559250,         62370;
  7:      6227020800,     15256200960,     10534043520,      3178375200,      472972500,      34054020,       945945;
  8:   1307674368000,   3390612825600,   2555413660800,    878139662400,   158918760000,   15664849200,    794593800,    16216200;
  9: 355687428096000, 966707617075200, 781533142790400, 296776697817600, 61886370546000, 7502781686400, 526815689400, 19848628800, 310134825;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(16*n^3-64*n^2+76*n-24)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-16*n+6+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*P[n - 1, x]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*v[i - 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 T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool 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.