A272867 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 42\,x + 99\,x^{2} + 96\,x^{3} + 33\,x^{4}\right)\,P_{n-1}(x)\)

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

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	30	33																											69
2	36	360	1296	1980	1089																									4761
3	216	3240	19764	62640	108702	98010	35937																							328509
4	1296	25920	222912	1075680	3183624	5916240	6743088	4312440	1185921																					22667121
5	7776	194400	2157840	13996800	58728240	166503600	323005320	423403200	359010630	177888150	39135393																			1564031349
6	46656	1399680	19035648	155131200	843482160	3222763200	8871171840	17725197600	25515335340	25809953400	17418807648	7044370740	1291467969																	107918163081
7	279936	9797760	157743936	1548046080	10343005344	49760373600	177744838320	478837526400	977596610760	1505251301400	1720817514108	1416558916080	793900582398	271208273490	42618442977															7446353252589
8	1679616	67184640	1249634304	14343920640	113704964352	659937922560	2900599414272	9847462636800	26096285121120	54161044502400	87743132281728	109797171865920	104047148942352	72190711343520	34590678081696	10228426314480	1406408618241													513798374428641
9	10077696	453496320	9568772352	125769646080	1153785337344	7844125847040	40953997986048	167880709693440	547721370501696	1432821087662400	3012467537759328	5078391468226560	6813721414928736	7177865407907040	5806821323893392	3481384374673920	1456783617839814	379730326925070	46411484401953											35452087835576229
10	60466176	3023308800	71350087680	1056646425600	11011570894080	85828713523200	519122679459840	2494760350694400	9674231447352960	30568631060908800	79134243082983936	168127470834998400	292645501282427040	415065753346780800	475029696873219840	431962504425777600	304807336635738420	160867574860984200	59744238102877680	13923445320585900	1531578985264449									2446194060654759801
11	362797056	19953838080	520795173888	8580150374400	100123371025920	880188740006400	6052979321959680	33374590034227200	150019247627124480	555965821358956800	1711114028087815296	4391556942509498880	9411127154482984128	16817966096108443200	24959452323962835360	30539835793195027200	30463699149404013240	24364188194288718600	15243134783426159940	7184497785422324400	2398452690924127134	505421065137268170	50542106513726817							168787390185178426269
12	2176782336	130606940160	3735358488576	67763234119680	875330978125824	8567978533171200	66027609375805440	410883148284364800	2101101504414232320	8937733828096972800	31896293980990703616	96029634602537026560	244680234129080775936	528162990313953646080	964862892924968784384	1487015465649633849600	1922639195383048462320	2067910684823288606400	1827686526459550954560	1304419240626846458400	732949097561277772824	312074534037484128240	94614823393696601424	18195158344941654120	1667889514952984961					11646329922777311412561
13	13060694016	848945111040	26402192953344	522950188400640	7408149722468352	79903987268751360	682090934528120832	4728870462575001600	27112014584417917440	130218591994392844800	528846128374523072256	1828011843372143370240	5401548144223288671744	13678362907639867031040	29708514793228087694592	55297358262007336949760	87986774608311276146592	119158150336869102544800	136450533151096563175920	130898163836716131398400	103843927177338241889424	66906755071749171081360	34117245181008869958936	13246075275117524199360	3678151259429955380358	650476910831664134790	55040353993448503713			803596764671634487466709
14	78364164096	5485491486720	184312513953792	3957782107668480	61008676354289664	718925771503779840	6734134166072721408	51472782391162429440	327047202223185540096	1750724714763869644800	7974331011364715053056	31130668447283516267520	104701566402863294522112	304441012603818593556480	766853064747933063266304	1674425569321002264560640	3167222383686614659293888	5179364962916795019008640	7297011271086924568237056	8811123938669874067646400	9052876072151699978093904	7836397752362274619318080	5638755720090690334545408	3310916724448693889649120	1545320256783398166907836	551367883326766944798360	141223541010099142617792	23116948677248371559460	1816331681783800622529	55448176762342779635202921

Technical details

Node id: n-1df5db9823996fe6 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-9217fe7884f21aef

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,        30,         33;
  2:       36,       360,       1296,         1980,          1089;
  3:      216,      3240,      19764,        62640,        108702,         98010,          35937;
  4:     1296,     25920,     222912,      1075680,       3183624,       5916240,        6743088,         4312440,         1185921;
  5:     7776,    194400,    2157840,     13996800,      58728240,     166503600,      323005320,       423403200,       359010630,        177888150,         39135393;
  6:    46656,   1399680,   19035648,    155131200,     843482160,    3222763200,     8871171840,     17725197600,     25515335340,      25809953400,      17418807648,       7044370740,       1291467969;
  7:   279936,   9797760,  157743936,   1548046080,   10343005344,   49760373600,   177744838320,    478837526400,    977596610760,    1505251301400,    1720817514108,    1416558916080,     793900582398,     271208273490,      42618442977;
  8:  1679616,  67184640, 1249634304,  14343920640,  113704964352,  659937922560,  2900599414272,   9847462636800,  26096285121120,   54161044502400,   87743132281728,  109797171865920,  104047148942352,   72190711343520,   34590678081696,   10228426314480,    1406408618241;
  9: 10077696, 453496320, 9568772352, 125769646080, 1153785337344, 7844125847040, 40953997986048, 167880709693440, 547721370501696, 1432821087662400, 3012467537759328, 5078391468226560, 6813721414928736, 7177865407907040, 5806821323893392, 3481384374673920, 1456783617839814, 379730326925070, 46411484401953;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+42*x+99*x^2+96*x^3+33*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[((6 + 42*x + 99*x^2 + 96*x^3 + 33*x^4)*P[n - 1, x])/(1 + 2*x + 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] = ((6 + 42*x + 99*x^2 + 96*x^3 + 33*x^4)*v[i - 1])/(1 + 2*x + 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 [-1, 0]
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 basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272867 → basis A013609 → x ↦ x+1

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.