A253284 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	18	24													42
3	144	480	360												984
4	1200	7200	12600	6720											27720
5	10800	100800	302400	362880	151200										928080
6	105840	1411200	6350400	12700800	11642400	3991680									36202320
7	1128960	20321280	127008000	372556800	558835200	415134720	121080960								1616065920
8	13063680	304819200	2514758400	10059033600	21794572800	26153487360	16345929600	4151347200							81337011840
9	163296000	4790016000	50295168000	261534873600	762810048000	1307674368000	1307674368000	705729024000	158789030400						4559460192000
10	2195424000	79035264000	1027458432000	6712728422400	25172731584000	57537672192000	81511702272000	69867173376000	33186907353600	6704425728000					281802030048000
11	31614105600	1369944576000	21576627072000	172613016576000	805527410688000	2347537025433600	4401631922688000	5309905176576000	3982428882432000	1689515283456000	309744468633600				19041881372236800
12	485707622400	24932991283200	467493586560000	4487938430976000	25431651108864000	91553943991910400	217440616980787200	345143836477440000	362401028301312000	241600685534208000	92613596121446400	15543540607795200			1396709749840204800
13	7933224499200	475993469952000	10471856338944000	118681038508032000	801097009929216000	3479049871692595200	10147228792436736000	20294457584873472000	27904879179201024000	25931806914004992000	15559084148402995200	5440239212728320000	841941782922240000		110529421317733017600
14	137305808640000	9519869399040000	242756669675520000	3204388039716864000	25368071981091840000	130464370188472320000	456625295659653120000	1116195167168040960000	1925436663364870656000	2333862622260449280000	1944885518550374400000	1060846646482022400000	340986422083507200000	48962152914554880000	9387089732537637120000

Technical details

Node id: n-c2b64ab37daf9153 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3961bf8dfc4504b6

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:         2;
  2:        18,         24;
  3:       144,        480,         360;
  4:      1200,       7200,       12600,         6720;
  5:     10800,     100800,      302400,       362880,       151200;
  6:    105840,    1411200,     6350400,     12700800,     11642400,       3991680;
  7:   1128960,   20321280,   127008000,    372556800,    558835200,     415134720,     121080960;
  8:  13063680,  304819200,  2514758400,  10059033600,  21794572800,   26153487360,   16345929600,   4151347200;
  9: 163296000, 4790016000, 50295168000, 261534873600, 762810048000, 1307674368000, 1307674368000, 705729024000, 158789030400;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] + (2*n*x + 2*n*x^2)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 2*n + 1 + (2*n^2 + 2*n)*x)*v[i - 1] + (2*n*x + 2*n*x^2)*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) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] 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

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.