A103633 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(7 + 3\,x - 5\,n - 2\,n x\right)\,P_{n-1}(x) + \left(12 + 10\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	6	5	1													12
3	12	16	7	1												36
4	36	60	37	10	1											144
5	72	156	134	57	12	1										432
6	216	540	558	305	93	15	1									1728
7	432	1296	1656	1168	491	123	17	1								5184
8	1296	4320	6264	5160	2641	860	174	20	1							20736
9	2592	9936	16848	16584	10442	4361	1208	214	22	1						62208
10	7776	32400	60480	66600	47910	23525	7985	1850	280	25	1					248832
11	15552	72576	153360	193680	162420	94960	39495	11685	2410	330	27	1				746496
12	46656	233280	532656	734400	680940	447300	213445	74550	18915	3400	411	30	1			2985984
13	93312	513216	1298592	2001456	2096280	1575540	874190	362545	112380	25715	4222	471	32	1		8957952
14	279936	1632960	4408992	7302960	8290296	6822900	4198110	1961825	699685	189525	38381	5635	567	35	1	35831808

Technical details

Node id: n-5924e3ed408d8904 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-2de66f1207bf9389

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 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    2,    1;
  2:    6,    5,     1;
  3:   12,   16,     7,     1;
  4:   36,   60,    37,    10,     1;
  5:   72,  156,   134,    57,    12,    1;
  6:  216,  540,   558,   305,    93,   15,    1;
  7:  432, 1296,  1656,  1168,   491,  123,   17,   1;
  8: 1296, 4320,  6264,  5160,  2641,  860,  174,  20,  1;
  9: 2592, 9936, 16848, 16584, 10442, 4361, 1208, 214, 22, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(5*n-7+(2*n-3)*x)*P(n-1,x) + (12+10*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(5*n - 7 + (2*n - 3)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(5*n - 7 + (2*n - 3)*x)*v[i - 1] + (12 + 10*x + 2*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 - 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 shift_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.