A272866 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 9\,x^{2} + 15\,n x + 18\,n x^{2}\right)\,P_{n}(x) = \left(25 + 150\,x + 315\,x^{2} + 270\,x^{3} + 81\,x^{4} + 75\,n x + 315\,n x^{2} + 405\,n x^{3} + 162\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	5	30	27																											62
2	25	300	945	1080	405																									2755
3	125	2250	12150	29700	36450	21870	5103																							107648
4	625	15000	114750	432000	921375	1166400	867510	349920	59049																					3926629
5	3125	93750	928125	4725000	14428125	28248750	36358875	30618000	16238475	4920750	649539																			137212514
6	15625	562500	6834375	43875000	174656250	464737500	857941875	1115370000	1018595250	639697500	263063295	63772920	6908733																	4656030823
7	78125	3281250	47250000	366187500	1798453125	6059812500	14623284375	25861275000	33842458125	32722987500	23074872975	11532269700	3868890480	781218270	71744535															154654063460
8	390625	18750000	312187500	2835000000	16567031250	67614750000	201823650000	452709000000	774833596875	1018595250000	1027570698000	788658444000	452176574850	187492384800	53090955900	9183300480	731794257													5054213758537
9	1953125	105468750	1993359375	20756250000	140674218750	674552812500	2401120968750	6534756000000	13849020309375	23099845781250	30467844680625	31758914160000	26006198195250	16522766410500	7974405065250	2823864897600	691545572865	104603532030	7360989291											163080330625286
10	9765625	585937500	12392578125	145546875000	1122451171875	6185109375000	25582774218750	81930487500000	207303508828125	419832239062500	685825902196875	906837636375000	970180421315625	836182555425000	575480851368750	311658260040000	129802006335375	40098020611500	8649162416925	1162261467000	73222472421									5208065405336971
11	48828125	3222656250	75410156250	986132812500	8527148437500	53094550781250	250296169921875	922435593750000	2715123765234375	6475937660156250	12636700052625000	20290982036062500	26881707384675000	29374853226468750	26391002998078125	19366719777900000	11485968235025625	5417591263053750	1983786613924500	543357235822500	104708135562030	12657027375630	721764371007							164917235453678792
12	244140625	17578125000	450878906250	6496875000000	62256884765625	432274218750000	2288077558593750	9540104062500000	32038003722656250	88009151484375000	199902833073281250	378177401407500000	598545531932765625	794172542955750000	883207062487406250	821230994331000000	635275168215806250	405600195446325000	211232788941858750	88215645345300000	28814873459474025	7087935330352800	1234217074421970	135566177510880	7060738412025					5185216650424977325
13	1220703125	95214843750	2656494140625	41818359375000	439864013671875	3369702832031250	19789623369140625	92111997656250000	347655600703125000	1081338656835937500	2804083205807812500	6112523111540625000	11264349657042421875	17610537868471406250	23395187749241953125	26406099841855500000	25273456850600887500	20432629231201687500	13868287848976275000	7832334743429625000	3634681155696966375	1361579719933397250	401355122607717525	89586648971773200	14227387900230375	1431917749958670	68630377364883			162047170708104820778
14	6103515625	512695312500	15457763671875	263935546875000	3023261718750000	25330614257812500	163460202978515625	840247108593750000	3521662177236328125	12237213269531250000	35686715474566406250	88126070964750000000	185475497491339453125	334196753396526562500	516978378949995703125	687490464129997500000	785845415524844390625	770926868799633000000	647079311177305256250	462460932065632500000	279485646590475950625	141464430720572602500	59186392612955141625	20098250675060922000	5397228442151910000	1102576667468175900	160938234920650635	14946171070574520	663426981193869	5037968698062673674799

Technical details

Node id: n-cdeab012a595d9e0 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f80803402873efeb

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:       5,        30,         27;
  2:      25,       300,        945,        1080,          405;
  3:     125,      2250,      12150,       29700,        36450,        21870,          5103;
  4:     625,     15000,     114750,      432000,       921375,      1166400,        867510,        349920,          59049;
  5:    3125,     93750,     928125,     4725000,     14428125,     28248750,      36358875,      30618000,       16238475,        4920750,         649539;
  6:   15625,    562500,    6834375,    43875000,    174656250,    464737500,     857941875,    1115370000,     1018595250,      639697500,      263063295,       63772920,        6908733;
  7:   78125,   3281250,   47250000,   366187500,   1798453125,   6059812500,   14623284375,   25861275000,    33842458125,    32722987500,    23074872975,    11532269700,     3868890480,      781218270,      71744535;
  8:  390625,  18750000,  312187500,  2835000000,  16567031250,  67614750000,  201823650000,  452709000000,   774833596875,  1018595250000,  1027570698000,   788658444000,   452176574850,   187492384800,   53090955900,    9183300480,    731794257;
  9: 1953125, 105468750, 1993359375, 20756250000, 140674218750, 674552812500, 2401120968750, 6534756000000, 13849020309375, 23099845781250, 30467844680625, 31758914160000, 26006198195250, 16522766410500, 7974405065250, 2823864897600, 691545572865, 104603532030, 7360989291;

%F Formula

The row polynomials satisfy (5+15*n*x+(18*n-9)*x^2)*P(n,x) = (25+(75*n+150)*x+(315*n+315)*x^2+(405*n+270)*x^3+(162*n+81)*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[((25 + (75*n + 150)*x + (315*n + 315)*x^2 + (405*n + 270)*x^3 + (162*n + 81)*x^4)*P[n - 1, x])/(5 + 15*n*x + (18*n - 9)*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] = ((25 + (75*n + 150)*x + (315*n + 315)*x^2 + (405*n + 270)*x^3 + (162*n + 81)*x^4)*v[i - 1])/(5 + 15*n*x + (18*n - 9)*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_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.