A280580 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-n + 2\,n^{2}\right)\,P_{n}(x) = \left(30\,x - n - 60\,n x + 2\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 12\,x^{2} + 4\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	1	6	2						9
4	1	15	30	5					51
5	1	28	140	140	14				323
6	1	45	420	1050	630	42			2188
7	1	66	990	4620	6930	2772	132		15511
8	1	91	2002	15015	42042	42042	12012	429	113634

All 15 rows

Technical details

Node id: n-e103f8fc18e5e16f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-d5ad2f0a26762abe

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   6,    2;
   4: 1,  15,   30,     5;
   5: 1,  28,  140,   140,     14;
   6: 1,  45,  420,  1050,    630,      42;
   7: 1,  66,  990,  4620,   6930,    2772,     132;
   8: 1,  91, 2002, 15015,  42042,   42042,   12012,     429;
   9: 1, 120, 3640, 40040, 180180,  336336,  240240,   51480,   1430;
  10: 1, 153, 6120, 92820, 612612, 1837836, 2450448, 1312740, 218790, 4862;

%F Formula

The row polynomials satisfy (2*n^2-n)*P(n,x) = (2*n^2-n+(24*n^2-60*n+30)*x)*P(n-1,x) + ((4*n-3)*x-(16*n-12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + (24*n^2 - 60*n + 30)*x)*P[n - 1, x] + ((4*n - 3)*x - (16*n - 12)*x^2)*D[P[n - 1, x], x])/(2*n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - n + (24*n^2 - 60*n + 30)*x)*v[i - 1] + ((4*n - 3)*x - (16*n - 12)*x^2)*deriv(v[i - 1], x))/(2*n^2 - n)); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = binomial(2*n,2*k)*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

Related ?

Same fingerprint ?

A280580: Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.

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.