A094262 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 4\,x + x^{2}\right)\,P_{n-1}(x) + \left(4 + 8\,x + 5\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

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	4	4	1																											9
2	32	72	60	22	3																									189
3	416	1472	2144	1648	706	160	15																							6561
4	7552	36480	76320	90400	66360	30936	8950	1470	105																					318573
5	176128	1078528	2947072	4734400	4954240	3530224	1735504	581476	127120	16380	945																			19882017
6	5018624	37223424	125637120	255247104	347751936	334821312	233675232	119144880	44059932	11527768	2026080	214830	10395																	1516358637
7	168968192	1471852544	5915737088	14544316416	24442102272	29707378176	26935746432	18512496384	9693451968	3848949616	1141043260	244949320	36001350	3243240	135135															136666371393
8	6563282944	65664000000	306262179840	884078520320	1768152468480	2598370274304	2902733191680	2514999893760	1708254504000	912773758880	382465002816	124369234440	30772269220	5601315720	707476770	55405350	2027025													14211824805549
9	288909131776	3264318570496	17331725123584	57463636582400	133337172776960	229965861994496	305555623289856	319847287916544	267425289790080	179952603113600	97699511273024	42699886231904	14913395202376	4106416840240	871921910860	137771694060	15256741500	1056755700	34459425											1674877679398881
10	14212910809088	178981260361728	1065865917726720	3991549377609728	10543259730296832	20881779579691008	32180312334460928	39517132912097280	39276147009589248	31909869341650432	21310059101814912	11719532142928512	5298861107615680	1959176573488992	586626142910808	140072178686440	26048541759240	3636340267560	358515857700	22260788550	654729075									220603503935140461
11	772776684683264	10731554109390848	70840955139915776	295735623626719232	876229977812271104	1960130228571029504	3438143869393215488	4847285975882510336	5585963662864504832	5321839965756734464	4222773299668481536	2802384175229485312	1557626339135741824	724203630542362048	280521341268584416	89874863433955600	23550772910304640	4964133145972000	821211660589320	102679676779680	9122994931050	513307594800	13749310575							32114506686565067649
12	46017323176296448	698597397170749440	5063003349988147200	23313513586722078720	76578383486989762560	190973644884563853312	375732415050778705920	598245668695410278400	784542927987199180800	857931458357349376000	789061692170110334976	613833566239293603840	405212395336669939200	227239883017057128960	108152160309380753280	43549162613538306816	14754197408397567120	4170368619407390400	971513970669161920	183296161895006160	27307598559040800	3091883854059000	250008297288750	12860188491150	316234143225					5120284528632900644397
13	2978458881388183552	49071264858659356672	387355623938140930048	1950311757914229964800	7034440471420592455680	19351553594174141104128	42210091925199281061888	74920857798348849610752	110196159588470573629440	136071112702963012157440	142389386868213098758144	127111529030934620483584	97233572472543500937216	63898024439585891466240	36105913420254116897280	17528983023349169889024	7293805375138418944704	2589939796727222861376	779698739033072818960	197167757532578472640	41343249877743333520	7059718241099924400	956659540685448600	98970464594301300	7342535160160200	347857557547500	7905853580625			887351420405903254279713
14	208198894960190160896	3699614005334592454656	31595473416342900572160	172680898029179210039296	678488919126808942608384	2041138382694196092862464	4889076055052283458617344	9572781021753809735516160	15609283006934667546722304	21484977924433208910872576	25212865733032446355341312	25411210167441210583990272	22111917986488303144837120	16671204777623506330386432	10913025066624919776780288	6206807374157163009626112	3065276575844118629966592	1311873260637406353137280	484879496235446697849152	153976227975500564900448	41707193349555952698480	9541510714986382612480	1819021535778558231120	283684319661443500800	35254577369633321700	3357550660635171000	230113044886725000	10103680876038750	213458046676875	166080356422879009442217453

Technical details

Node id: n-5a364d1118d1dc6e · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-982865a374cd90cf

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:            4,             4,              1;
  2:           32,            72,             60,             22,               3;
  3:          416,          1472,           2144,           1648,             706,             160,              15;
  4:         7552,         36480,          76320,          90400,           66360,           30936,            8950,            1470,             105;
  5:       176128,       1078528,        2947072,        4734400,         4954240,         3530224,         1735504,          581476,          127120,           16380,            945;
  6:      5018624,      37223424,      125637120,      255247104,       347751936,       334821312,       233675232,       119144880,        44059932,        11527768,        2026080,         214830,          10395;
  7:    168968192,    1471852544,     5915737088,    14544316416,     24442102272,     29707378176,     26935746432,     18512496384,      9693451968,      3848949616,     1141043260,      244949320,       36001350,       3243240,       135135;
  8:   6563282944,   65664000000,   306262179840,   884078520320,   1768152468480,   2598370274304,   2902733191680,   2514999893760,   1708254504000,    912773758880,   382465002816,   124369234440,    30772269220,    5601315720,    707476770,     55405350,     2027025;
  9: 288909131776, 3264318570496, 17331725123584, 57463636582400, 133337172776960, 229965861994496, 305555623289856, 319847287916544, 267425289790080, 179952603113600, 97699511273024, 42699886231904, 14913395202376, 4106416840240, 871921910860, 137771694060, 15256741500, 1056755700, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (4+4*x+x^2)*P(n-1,x) + (4+8*x+5*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[(4 + 4*x + x^2)*P[n - 1, x] + (4 + 8*x + 5*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] = (4 + 4*x + x^2)*v[i - 1] + (4 + 8*x + 5*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 (−∞, -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 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.