basis-map coefficients of A371080, shifted → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 4\,x + 3\,n\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	4														9
2	40	68	16													124
3	440	1180	576	64												2260
4	6160	23000	17392	3968	256											50776
5	104720	507640	526800	184640	24320	1024										1349144
6	2094400	12602240	16780960	8015680	1614080	138240	4096									41249696
7	48171200	348638080	570618720	347672640	95011840	12400640	745472	16384								1423274976
8	1252451200	10651827200	20795588800	15494035200	5381187840	950476800	86876160	3866624	65536							54616375360
9	36321084800	356520102400	812044094400	718437798400	304129593600	68098114560	8406343680	567902208	19464192	262144						2304544760384
10	1162274713600	12980008025600	33907844185600	34859439507200	17471971686400	4757620331520	743147704320	67699507200	3517317120	95682560	1048576					105953619709696
11	40679614976000	510869411852800	1509957332083200	1774025033580800	1030508314035200	331557004979200	62876195880960	7237659770880	506458275840	20862730240	461373440	4194304				5268238353732864
12	1545825369088000	21619233757721600	71501514923238400	94741081007372800	62743549092224000	23352339544934400	5224552164536320	729190328401920	64402718392320	3569675141120	119438049280	2189426688	16777216			281525380225304064
13	63378840132608000	979048820573824000	3590051166269568000	5307283353083712000	3955346622286259200	1675466908609894400	433005248874598400	71212341317877760	7618159756247040	532475859435520	23953182556160	663853465600	10250878976	67108864		16082968564118033920
14	2788668965834752000	47247858748073984000	190598855928312832000	311568072437766144000	258549930749510400000	123051268640178278400	36146224557912268800	6859309570385592320	863829506746286080	73070707779829760	4141970772459520	154231835197440	3598494728192	47445966848	268435456	977751389661317154816

Technical details

Node id: n-250be2442d9d6c70 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-fa42e2e61b188dff

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:           1;
  1:           5,            4;
  2:          40,           68,           16;
  3:         440,         1180,          576,           64;
  4:        6160,        23000,        17392,         3968,          256;
  5:      104720,       507640,       526800,       184640,        24320,        1024;
  6:     2094400,     12602240,     16780960,      8015680,      1614080,      138240,       4096;
  7:    48171200,    348638080,    570618720,    347672640,     95011840,    12400640,     745472,     16384;
  8:  1252451200,  10651827200,  20795588800,  15494035200,   5381187840,   950476800,   86876160,   3866624,    65536;
  9: 36321084800, 356520102400, 812044094400, 718437798400, 304129593600, 68098114560, 8406343680, 567902208, 19464192, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+2+4*x)*P(n-1,x) + 4*x*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[(3*n + 2 + 4*x)*P[n - 1, x] + 4*x*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] = (3*n + 2 + 4*x)*v[i - 1] + 4*x*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 (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from basis_map_A013611 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

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