A094262 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	4	3																											8
2	1	12	36	40	15																									104
3	1	28	183	496	655	420	105																							1888
4	1	60	720	3560	9150	13356	11200	5040	945																					44032
5	1	124	2523	20240	84950	211296	329434	326368	200025	69300	10395																			1254656
6	1	252	8316	101640	639765	2408616	5858832	9572256	10639755	7957180	3839220	1081080	135135																	42242048
7	1	508	26463	474096	4256805	22738716	79210803	189588288	320787891	388089460	334326685	200600400	79774695	18918900	2027025															1640820736
8	1	1020	82440	2110120	26157300	190358532	902615280	2965988880	7012577430	12192535300	15744228456	15088118760	10598928340	5306961660	1794592800	367567200	34459425													72227282944
9	1	2044	253443	9104080	152299900	1468699512	9166427508	39692623776	124624534230	291836790400	518560641346	705229331664	734752126108	582301119400	345111666900	148219431360	43591172625	7856748900	654729075											3553227702272
10	1	4092	772596	38452040	853807515	10694420352	85792885024	475953426432	1918585883610	5807784826360	13495446320136	24410248857072	34629028600166	38598485845920	33666990477120	22731902152960	11647408938165	4378653459180	1139228590500	183324141000	13749310575									193194171170816
11	1	8188	2342343	160005296	4657420955	74642924532	756248732701	5265868934656	26606570593482	101209100200120	297325620421126	686507115884448	1260284644247798	1851735086646536	2182394861536890	2058629903249920	1543229463260485	907296722792460	409334944493475	136808695623600	31912149844575	4638100767300	316234143225							11504330794074112
12	1	16380	7076160	658792680	24898869450	504706630428	6371344019040	54824074596720	341051014759455	1596881103217400	5789854305563328	16588409483121840	38098118453691660	70823360874372120	107192245186166400	132402475407523776	133329030232052655	108942939946140300	71575999213438720	37255506148060200	15012154197290250	4515964113761100	954605467015200	126493657290000	7905853580625					744614720347045888
13	1	32764	21326763	2691845520	131117654850	3330978972048	51839593045422	544571754121824	4113634571747415	23338072073330300	102571342794333413	357179990369590752	1001842292431482684	2290512101708281920	4304056280829034500	6681765226467659136	8590545845965532319	9144062564719848948	8033027277417667525	5787008890421558800	3383082479843836770	1579507130440839600	574969821512035950	157240048921956000	30387465879395625	3699939475732500	213458046676875			52049723740047540224
14	1	65532	64176876	10938127240	682613363265	21575436763608	410428680510576	5212655353477728	47291559057328365	321099129709469540	1686972336640155708	7027928976379054536	23644057007544968813	65119273477171732560	148303841650156215360	281286238237344160896	446388253855408046547	594124550696134308996	663322266803635057060	619942520199534855960	482792678300197059795	310918981572101582680	163692530931463694160	69280483737284820000	22995934630883589375	5764587924068473500	1026021677693512500	115551955934415000	6190283353629375	3907812900294782615552

Technical details

Node id: n-49a8d5132607d3b9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-532b1e88c6581998

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,    4,      3;
  2: 1,   12,     36,      40,        15;
  3: 1,   28,    183,     496,       655,        420,        105;
  4: 1,   60,    720,    3560,      9150,      13356,      11200,        5040,          945;
  5: 1,  124,   2523,   20240,     84950,     211296,     329434,      326368,       200025,        69300,        10395;
  6: 1,  252,   8316,  101640,    639765,    2408616,    5858832,     9572256,     10639755,      7957180,      3839220,      1081080,       135135;
  7: 1,  508,  26463,  474096,   4256805,   22738716,   79210803,   189588288,    320787891,    388089460,    334326685,    200600400,     79774695,     18918900,      2027025;
  8: 1, 1020,  82440, 2110120,  26157300,  190358532,  902615280,  2965988880,   7012577430,  12192535300,  15744228456,  15088118760,  10598928340,   5306961660,   1794592800,    367567200,    34459425;
  9: 1, 2044, 253443, 9104080, 152299900, 1468699512, 9166427508, 39692623776, 124624534230, 291836790400, 518560641346, 705229331664, 734752126108, 582301119400, 345111666900, 148219431360, 43591172625, 7856748900, 654729075;

%F Formula

The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-1,x) + (x+2*x^2+x^3)*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[(1 + 4*x + 3*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*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] = (1 + 4*x + 3*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.