A272866 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(16 + 20\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)

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

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	16	20	5																											41
2	256	640	560	200	25																									1681
3	4096	15360	23040	17600	7200	1500	125																							68921
4	65536	327680	696320	819200	582400	256000	68000	10000	625																					2825761
5	1048576	6553600	18022400	28672000	29184000	19840000	9120000	2800000	550000	62500	3125																			115856201
6	16777216	125829120	424673280	851968000	1130496000	1044480000	688640000	326400000	110400000	26000000	4050000	375000	15625																	4750104241
7	268435456	2348810240	9395240960	22754099200	37250662400	43581440000	37560320000	24217600000	11737600000	4256000000	1136800000	217000000	28000000	2187500	78125															194754273881
8	4294967296	42949672960	198642237440	563714457600	1098068787200	1556086784000	1658847232000	1356595200000	859955200000	423936000000	161996800000	47488000000	10472000000	1680000000	185000000	12500000	390625													7984925229121
9	68719476736	773094113280	4058744094720	13207024435200	29836600934400	49677336576000	63153635328000	62662901760000	49185423360000	30765056000000	15370444800000	6119424000000	1927296000000	473760000000	88920000000	12300000000	1181250000	70312500	1953125											327381934393961
10	1099511627776	13743895347200	80745385164800	296352743424000	761819824128000	1457604526080000	2153187901440000	2514065817600000	2355992985600000	1789263872000000	1107155353600000	559144960000000	230077440000000	76723200000000	20534400000000	4344000000000	709500000000	86250000000	7343750000	390625000	9765625									13422659310152401
11	17592186044416	241892558110720	1572301627719680	6425271074816000	18519898980352000	40039832616960000	67412196065280000	90576833740800000	98743143628800000	88318410752000000	65279728025600000	40035745792000000	20399915008000000	8624844800000000	3013401600000000	863808000000000	200904000000000	37290000000000	5390000000000	584375000000	44687500000	2148437500	48828125							550329031716248441
12	281474976710656	4222124650659840	30082638135951360	135459832542003200	432685313320550400	1043161656852480000	1971991284285440000	2997672424243200000	3728487953203200000	3840841613312000000	3304559581593600000	2387758546944000000	1453512196096000000	746174545920000000	322710896640000000	117213184000000000	35557632000000000	8933760000000000	1836560000000000	303600000000000	39352500000000	3850000000000	267187500000	11718750000	244140625					22563490300366186081
13	4503599627370496	73183493944770560	567172078071971840	2790120706644377600	9782574854648627200	26021591938760704000	54578520252219392000	92618392757862400000	129469212157542400000	151011586998272000000	148332225337753600000	123499536449536000000	87534163984384000000	52947829391360000000	27354426245120000000	12060501606400000000	4526740275200000000	1440158720000000000	385848320000000000	86257600000000000	15884440000000000	2366650000000000	278037500000000	24781250000000	1574218750000	63476562500	1220703125			925103102315013629321
14	72057594037927936	1261007895663738880	10560941126183813120	56351290337473331200	215159472197625446400	625947571646365696000	1442601037083967488000	2703572648863989760000	4196767850939023360000	5468661598912512000000	6040893406157209600000	5697695024939008000000	4612198479429632000000	3215346232197120000000	1934298227671040000000	1004795697561600000000	450410007756800000000	173879853056000000000	57610448896000000000	16297881600000000000	3908544640000000000	786843200000000000	131203800000000000	17790500000000000	1911000000000000	156406250000000	9160156250000	341796875000	6103515625	37929227194915558802161

Technical details

Node id: n-bc4e6b211c5c1593 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-d98c1510b99a0b49

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:          16,           20,             5;
  2:         256,          640,           560,            200,             25;
  3:        4096,        15360,         23040,          17600,           7200,           1500,            125;
  4:       65536,       327680,        696320,         819200,         582400,         256000,          68000,          10000,            625;
  5:     1048576,      6553600,      18022400,       28672000,       29184000,       19840000,        9120000,        2800000,         550000,          62500,           3125;
  6:    16777216,    125829120,     424673280,      851968000,     1130496000,     1044480000,      688640000,      326400000,      110400000,       26000000,        4050000,        375000,         15625;
  7:   268435456,   2348810240,    9395240960,    22754099200,    37250662400,    43581440000,    37560320000,    24217600000,    11737600000,     4256000000,     1136800000,     217000000,      28000000,      2187500,       78125;
  8:  4294967296,  42949672960,  198642237440,   563714457600,  1098068787200,  1556086784000,  1658847232000,  1356595200000,   859955200000,   423936000000,   161996800000,   47488000000,   10472000000,   1680000000,   185000000,    12500000,     390625;
  9: 68719476736, 773094113280, 4058744094720, 13207024435200, 29836600934400, 49677336576000, 63153635328000, 62662901760000, 49185423360000, 30765056000000, 15370444800000, 6119424000000, 1927296000000, 473760000000, 88920000000, 12300000000, 1181250000, 70312500, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (16+20*x+5*x^2)*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[(16 + 20*x + 5*x^2)*P[n - 1, 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] = (16 + 20*x + 5*x^2)*v[i - 1]); 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 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_A013611 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)

Related ?

Same fingerprint ?

A272866 → basis A013611

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.