A144442 → γ

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(1 - 10\,x + 10\,n x\right)\,P_{n-1}(x) + \left(5\,x - 20\,x^{2}\right)\,P_{n-1}'(x) + \left(5\,x\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	15							16
3	1	115							116
4	1	725	1225						1951
5	1	4395	28550						32946
6	1	26425	449525	291625					767576
7	1	158615	6023750	13799250					19981616
8	1	951765	74324125	403748125	139450625				618474641
9	1	5710675	875464350	9463053750	11072421875				21416650651
10	1	34264145	10034613925	195553698125	518431212500	111421471875			835475260571
11	1	205584975	113150438150	3735313327250	18756518656250	13320944628125			35926132634751
12	1	1233509965	1263328788125	67735716976125	581630326634375	911632276081250	133766553640625		1696029435630466
13	1	7401059915	14020995590750	1185403526858750	16297056444525625	47061434836768750	22446013407625000		87003936612428791
14	1	44406359625	155051235638725	20234662676848125	425555110800031250	2041358279615440625	2112227622119843750	225128966844453125	4824659737698615226

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Technical details

Node id: n-6eb71148be3afc89 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-4a72b1b0c69858a0

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,      15;
  3: 1,     115;
  4: 1,     725,      1225;
  5: 1,    4395,     28550;
  6: 1,   26425,    449525,     291625;
  7: 1,  158615,   6023750,   13799250;
  8: 1,  951765,  74324125,  403748125,   139450625;
  9: 1, 5710675, 875464350, 9463053750, 11072421875;

%F Formula

The row polynomials satisfy P(n,x) = (1+(10*n-10)*x)*P(n-1,x) + (5*x-20*x^2)*d/dx P(n-1,x) + 5*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (10*n - 10)*x)*P[n - 1, x] + (5*x - 20*x^2)*D[P[n - 1, x], x] + 5*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + (10*n - 10)*x)*v[i - 1] + (5*x - 20*x^2)*deriv(v[i - 1], x) + 5*x*v[i - 2]); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-a34157123148cce9
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-a34157123148cce9
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-a34157123148cce9

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

gamma_polynomial scale 1, shift 0 e-a34157123148cce9

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-0086426c09dadf26 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

No candidate equalities or related sequences.

References

[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