A200536 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 4\,x^{2} + 7\,n x + 8\,n x^{2}\right)\,P_{n}(x) = \left(18 + 84\,x + 146\,x^{2} + 112\,x^{3} + 32\,x^{4} + 42\,n x + 146\,n x^{2} + 168\,n x^{3} + 64\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	6	28	24																											58
2	36	336	876	896	320																									2464
3	216	3024	13176	27104	29280	16128	3584																							92512
4	1296	24192	147744	456960	825680	913920	612864	229376	36864																					3248896
5	7776	181440	1425600	5886720	14920800	24838464	27852160	20930560	10137600	2867200	360448																			109408768
6	46656	1306368	12550464	64834560	211550400	470340864	739920832	836161536	676961280	384204800	145440768	33030144	3407872																	3579756544
7	279936	9144576	103856256	644184576	2571246720	7154106624	14491123584	21872671744	24841926144	21197352960	13408575488	6107824128	1896480768	359661568	31457280															114689892352
8	1679616	62705664	821892096	5950070784	28085460480	93851799552	232595112960	439404478464	642597693696	732340797440	649789521920	444927049728	230786203648	87757422592	23089643520	3758096384	285212672													3616104841216
9	10077696	423263232	6288482304	52045701120	283597562880	1106347382784	3243040561152	7359034540032	13168613574144	18784053693440	21459962861568	19624092106752	14276242046976	8158742839296	3585233387520	1169707499520	266959060992	38050725888	2550136832											112584995504128
10	60466176	2821754880	46861286400	436431421440	2696231577600	12027071766528	40704876656640	107843894968320	228273464079360	391139767603200	546793981217792	625823628165120	586183710228480	447352453201920	275673238732800	135151950495744	51507269468160	14712947343360	2964534067200	375809638400	22548578304									3469733552717824
11	362797056	18623582208	341875759104	3538480619520	24439756492800	122696298160128	470869225795584	1426630161825792	3486860861552640	6976198107648000	11536323512727552	15860918887489536	18178449171570688	17362981956812800	13775252786380800	9017711640182784	4818866244943872	2068164762402816	695747456532480	176724482457600	31872415432704	3637837299712	197568495616							106038442476961792
12	2176782336	121899810816	2451056910336	27910541869056	213120293253120	1189227465474048	5102980164108288	17402449097392128	48233816540540928	110363548110520320	210745142580436992	338328137898196992	458698659852439552	526288214508306432	510897315346513920	418563975056195584	287947283380568064	165000998849347584	77823697490018304	29697154083717120	8940956631957504	2044464562438144	333693189095424	34634616274944	1717986918400					3217881673379086336
13	13060694016	792348770304	17318480265216	215166710513664	1799837001031680	11050712221925376	52428876871827456	198747470175141888	615979268500856832	1586509592489902080	3435689035892146176	6308111975933214720	9876001580422029312	13231222984182611968	15193848585264660480	14952561720730583040	12585958488386043904	9025476792831442944	5479877753979273216	2791734011788001280	1178319229997285376	404735773913382912	110265496401608704	22925229755990016	3417075980697600	325180563914752	14843406974976			97067234807290789888
14	78364164096	5119792054272	120863662424064	1626956141690880	14795352510382080	99123535475048448	515226401933967360	2149221114700627968	7365416714088136704	21088431988926382080	51069290342816858112	105551505582582398976	187411919275045011456	287162765816066015232	380794527785915105280	437581357434005356544	435692667032583274496	375294242071404085248	278789032802986754048	177732957667988602880	96562061283189325824	44277689320508227584	16909773814408151040	5280625937828806656	1313659126651289600	250389034214359040	34340222036606976	3017059906617344	127543348822016	2912945917279080546304

Technical details

Node id: n-9df7bf46a00d9e2f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7c392c1c35db3029

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:        6,        28,         24;
  2:       36,       336,        876,         896,          320;
  3:      216,      3024,      13176,       27104,        29280,         16128,          3584;
  4:     1296,     24192,     147744,      456960,       825680,        913920,        612864,        229376,          36864;
  5:     7776,    181440,    1425600,     5886720,     14920800,      24838464,      27852160,      20930560,       10137600,        2867200,         360448;
  6:    46656,   1306368,   12550464,    64834560,    211550400,     470340864,     739920832,     836161536,      676961280,      384204800,      145440768,       33030144,        3407872;
  7:   279936,   9144576,  103856256,   644184576,   2571246720,    7154106624,   14491123584,   21872671744,    24841926144,    21197352960,    13408575488,     6107824128,     1896480768,     359661568,      31457280;
  8:  1679616,  62705664,  821892096,  5950070784,  28085460480,   93851799552,  232595112960,  439404478464,   642597693696,   732340797440,   649789521920,   444927049728,   230786203648,   87757422592,   23089643520,    3758096384,    285212672;
  9: 10077696, 423263232, 6288482304, 52045701120, 283597562880, 1106347382784, 3243040561152, 7359034540032, 13168613574144, 18784053693440, 21459962861568, 19624092106752, 14276242046976, 8158742839296, 3585233387520, 1169707499520, 266959060992, 38050725888, 2550136832;

%F Formula

The row polynomials satisfy (3+7*n*x+(8*n-4)*x^2)*P(n,x) = (18+(42*n+84)*x+(146*n+146)*x^2+(168*n+112)*x^3+(64*n+32)*x^4)*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[Together[((18 + (42*n + 84)*x + (146*n + 146)*x^2 + (168*n + 112)*x^3 + (64*n + 32)*x^4)*P[n - 1, x])/(3 + 7*n*x + (8*n - 4)*x^2)]];
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] = ((18 + (42*n + 84)*x + (146*n + 146)*x^2 + (168*n + 112)*x^3 + (64*n + 32)*x^4)*v[i - 1])/(3 + 7*n*x + (8*n - 4)*x^2)); 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 basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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.