A225473 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−2, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(8 + 12\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	7	4							11
3	81	104	32						217
4	1399	2812	1824	384					6419
5	32289	88208	88256	38400	6144				253297
6	931687	3217204	4363200	2906880	952320	122880			12494171
7	32259441	134664824	230761952	207828480	103802880	27279360	2949120		739546057
8	1303134679	6379860652	13218822624	15027460224	10125588480	4045332480	887685120	82575360	51070459619

All 15 rows

Technical details

Node id: n-f9bf0dcf699ce3e0 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-5638cd0be2cbc35d

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:             7,              4;
   3:            81,            104,             32;
   4:          1399,           2812,           1824,            384;
   5:         32289,          88208,          88256,          38400,           6144;
   6:        931687,        3217204,        4363200,        2906880,         952320,         122880;
   7:      32259441,      134664824,      230761952,      207828480,      103802880,       27279360,        2949120;
   8:    1303134679,     6379860652,    13218822624,    15027460224,    10125588480,     4045332480,      887685120,      82575360;
   9:   60160827969,   337931053088,   821481431936,  1128825492480,   959160029184,   516157931520,   171839324160,   32369541120,    2642411520;
  10: 3124574220487, 19805036231524, 55261184634240, 89090394293760, 91461326745600, 62014069555200, 27775791267840, 7926243655680, 1307993702400, 95126814720;

%F Formula

The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) + (8+12*x+4*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[(7 + 4*x)*P[n - 1, x] + (8 + 12*x + 4*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] = (7 + 4*x)*v[i - 1] + (8 + 12*x + 4*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 (-2, -1)
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741