A013612 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 30\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 256 + 960\,x + 900\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	256	960	900															2116
1	4096	23040	43200	27000														97336
2	65536	491520	1382400	1728000	810000													4477456
3	1048576	9830400	36864000	69120000	64800000	24300000												205962976
4	16777216	188743680	884736000	2211840000	3110400000	2332800000	729000000											9474296896
5	268435456	3523215360	19818086400	61931520000	116121600000	130636800000	81648000000	21870000000										435817657216
6	4294967296	64424509440	422785843200	1585446912000	3715891200000	5573836800000	5225472000000	2799360000000	656100000000									20047612231936
7	68719476736	1159641169920	8697308774400	38050725888000	107017666560000	200658124800000	250822656000000	201553920000000	94478400000000	19683000000000								922190162669056
8	1099511627776	20615843020800	173946175488000	869730877440000	2853804441600000	6421059993600000	10032906240000000	10749542400000000	7558272000000000	3149280000000000	590490000000000							42420747482776576
9	17592186044416	362838837166080	3401614098432000	19134079303680000	71752797388800000	188351093145600000	353158299648000000	472979865600000000	443418624000000000	277136640000000000	103926240000000000	17714700000000000						1951354384207722496
10	281474976710656	6333186975989760	65310990689894400	408193691811840000	1722067137331200000	5166201411993600000	11301065588736000000	18162426839040000000	21284093952000000000	17736744960000000000	9976919040000000000	3401222400000000000	531441000000000000					89762301673555234816
11	4503599627370496	109775240917155840	1234971460318003200	8490428789686272000	39798884951654400000	134321236711833600000	335803091779584000000	629630797086720000000	885418308403200000000	922310737920000000000	691733053440000000000	353727129600000000000	110539728000000000000	15943230000000000000				4129065876983540801536
12	72057594037927936	1891511843495608320	23052800592602726400	172896004444520448000	891495022917058560000	3343106335938969600000	9402486569828352000000	20148185506775040000000	33055616847052800000000	41319521058816000000000	38737050992640000000000	26411625676800000000000	12380449536000000000000	3571283520000000000000	478296900000000000000			189937030341242876870656
13	1152921504606846976	32425917317067571200	425590164786511872000	3457920088890408960000	19450800500008550400000	80234552062535270400000	250732975195422720000000	604445565203251200000000	1133335434756096000000000	1652780842352640000000000	1859378447646720000000000	1584697540608000000000000	990435962880000000000000	428554022400000000000000	114791256000000000000000	14348907000000000000000		8737103395697172336050176
14	18446744073709551616	553402322211286548480	7782220156096217088000	68094426365841899520000	414950410666849075200000	1867276848000820838400000	6418764165002821632000000	17193118299114700800000000	36266733912195072000000000	60444556520325120000000000	79333480432926720000000000	81136514079129600000000000	63387901624320000000000000	36569943244800000000000000	14693280768000000000000000	3673320192000000000000000	430467210000000000000000	401906756202069927458308096

Technical details

Node id: n-69d39bd7c8ec9d15 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-dd0a803b420669cd

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            256,             960,              900;
  1:           4096,           23040,            43200,             27000;
  2:          65536,          491520,          1382400,           1728000,            810000;
  3:        1048576,         9830400,         36864000,          69120000,          64800000,           24300000;
  4:       16777216,       188743680,        884736000,        2211840000,        3110400000,         2332800000,          729000000;
  5:      268435456,      3523215360,      19818086400,       61931520000,      116121600000,       130636800000,        81648000000,        21870000000;
  6:     4294967296,     64424509440,     422785843200,     1585446912000,     3715891200000,      5573836800000,      5225472000000,      2799360000000,       656100000000;
  7:    68719476736,   1159641169920,    8697308774400,    38050725888000,   107017666560000,    200658124800000,    250822656000000,    201553920000000,     94478400000000,     19683000000000;
  8:  1099511627776,  20615843020800,  173946175488000,   869730877440000,  2853804441600000,   6421059993600000,  10032906240000000,  10749542400000000,   7558272000000000,   3149280000000000,    590490000000000;
  9: 17592186044416, 362838837166080, 3401614098432000, 19134079303680000, 71752797388800000, 188351093145600000, 353158299648000000, 472979865600000000, 443418624000000000, 277136640000000000, 103926240000000000, 17714700000000000;

%F Formula

The row polynomials satisfy P(n,x) = (16+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 960*x + 900*x^2.

%t Mathematica

P[0, x_] := 256 + 960*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 30*x)*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] = 256 + 960*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 30*x)*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+5x)^n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    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

Related ?

Same fingerprint ?

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.