A049029 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	5														6
3	1	15	45													61
4	1	30	255	585												871
5	1	50	825	5175	9945											15996
6	1	75	2025	24150	123795	208845										358891
7	1	105	4200	80850	775845	3427515	5221125									9509641
8	1	140	7770	219450	3363045	27478710	108046575	151412625								290528316
9	1	180	13230	514710	11482695	148548330	1069801425	3824996175	4996616625							10051973371
10	1	225	21150	1083600	33100515	619338825	7011734625	45547251750	150346471275	184874815125						388433817091
11	1	275	32175	2098800	84029715	2141962515	34881981750	354063575250	2108878296525	6499426608675	7579867420125					16579346005806
12	1	330	47025	3804075	193167315	6427477980	141980107500	2063280681000	19103929908525	105616706545350	306553491419175	341094033905625				774580047063901
13	1	390	66495	6531525	409999590	17244847620	495491396400	9730206486000	128457806001525	1099021061788650	5693005525232025	15668768604864375	16713607661375625			39313104018590221
14	1	455	91455	10720710	814954140	42254822610	1530182253600	38964198873600	692809782189525	8421116003875575	67238184985396425	328784072492625750	862827112324051875	885821206052908125		2153825039102763846
15	1	525	122850	16939650	1533241710	96041795850	4276745723250	136895863104000	3147554311226325	51375322499626125	580926261221806500	4363075171616411250	20261087389388971125	50929793720847916875	50491808745015763125	126681355435102649161

Technical details

Node id: n-ef665de83611fdca · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-e050e5a67817afb8

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,   5;
   3: 1,  15,    45;
   4: 1,  30,   255,     585;
   5: 1,  50,   825,    5175,     9945;
   6: 1,  75,  2025,   24150,   123795,    208845;
   7: 1, 105,  4200,   80850,   775845,   3427515,    5221125;
   8: 1, 140,  7770,  219450,  3363045,  27478710,  108046575,   151412625;
   9: 1, 180, 13230,  514710, 11482695, 148548330, 1069801425,  3824996175,   4996616625;
  10: 1, 225, 21150, 1083600, 33100515, 619338825, 7011734625, 45547251750, 150346471275, 184874815125;

%F Formula

The row polynomials satisfy P(n,x) = (1+(5*n-5)*x)*P(n-1,x) - 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[(1 + (5*n - 5)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
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] = (1 + (5*n - 5)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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
[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.