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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	60160827969	337931053088	821481431936	1128825492480	959160029184	516157931520	171839324160	32369541120	2642411520							4030568042977
10	3124574220487	19805036231524	55261184634240	89090394293760	91461326745600	62014069555200	27775791267840	7926243655680	1307993702400	95126814720						357861741121451
11	180312309395601	1272972939428744	4009706476563872	7420783626101760	8936382061916160	7317407175966720	4126494824202240	1582721925120000	395212278988800	58027356979200	3805072588800					35303826047251897
12	11445969681199159	89063038711750492	312583491315352224	653174399622459264	902929838461685760	867065673589800960	590243291141898240	284863136118865920	95531936042188800	21205193903308800	2804338497945600	167423193907200				3831077730280362419
13	792626097462398049	6739317475293691568	26078778131408945216	60731255910323919360	94794558452944103424	104483674874561495040	83391131789021675520	48563401445490032640	20482183729187389440	6102094076234956800	1219091986435276800	146662717862707200	8036313307545600			453532812999534136657
14	59462922484586318887	548477986522972855444	2319906803723800898880	5957815212215047184640	10364747804459568199680	12899071711527542661120	11815058089269630074880	8065080980300192808960	4103044543850899046400	1536626791932588195840	411814764498596659200	74797651263592857600	8253293766849331200	417888291992371200		58164576444107859463931
15	4804064349575887075761	47777442293455643911064	219292499900782743140192	615696865512638822115840	1181349041101818712381440	1638655549046478965882880	1694611845025772597084160	1327338846513324726681600	791333308405395121766400	357366865810767740928000	120347740079821475020800	29309039365195196006400	4879995001813912780800	497287067470921728000	23401744351572787200	8033283791218664039290537

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