A281000 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 + 8\,x - 16\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 3\,x^{2} + 16\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 0\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	10	2													12
4	70	42	3												115
5	420	504	108	4											1036
6	2310	4620	1980	220	5										9135
7	12012	36036	25740	5720	390	6									79904
8	60060	252252	270270	100100	13650	630	7								696969
9	291720	1633632	2450448	1361360	309400	28560	952	8							6076080
10	1385670	9976824	19953648	15519504	5290740	813960	54264	1368	9						52995987
11	6466460	58198140	149652360	155195040	74070360	17093160	1899240	95760	1890	10					462672420
12	29745716	327202876	1051723530	1402298040	892371480	288304632	48050772	4037880	159390	2530	11				4043896857
13	135207800	1784742960	7011490200	11685817000	9561123000	4118637600	961015440	121136400	7969500	253000	3300	12			35387396212
14	608435100	9491587560	44746055640	91149372600	93220949250	51630064200	16062686640	2834591760	279729450	14800500	386100	4212	13		310038663025
15	2714556600	49404930120	275256039240	672848095920	841060119900	582272390700	232908956280	54802107360	7571343780	600900300	26126100	570024	5278	14	2719466141616

Technical details

Node id: n-543256a15ee14711 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e65c25f9531a60c5

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 -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       0;
   2:       1;
   3:      10,       2;
   4:      70,      42,        3;
   5:     420,     504,      108,        4;
   6:    2310,    4620,     1980,      220,       5;
   7:   12012,   36036,    25740,     5720,     390,      6;
   8:   60060,  252252,   270270,   100100,   13650,    630,     7;
   9:  291720, 1633632,  2450448,  1361360,  309400,  28560,   952,    8;
  10: 1385670, 9976824, 19953648, 15519504, 5290740, 813960, 54264, 1368, 9;

%F Formula

The row polynomials satisfy (2*n^2-3*n-2)*P(n,x) = (8*n^2-2+(6*n^2-16*n+8)*x)*P(n-1,x) + ((16*n-12)*x-(4*n-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 2 + (6*n^2 - 16*n + 8)*x)*P[n - 1, x] + ((16*n - 12)*x - (4*n - 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 2 + (6*n^2 - 16*n + 8)*x)*v[i - 1] + ((16*n - 12)*x - (4*n - 3)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.?
  - 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
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.