A038321 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 18\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 784 + 1008\,x + 324\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	784	1008	324															2116
1	21952	42336	27216	5832														97336
2	614656	1580544	1524096	653184	104976													4477456
3	17210368	55319040	71124480	45722880	14696640	1889568												205962976
4	481890304	1858719744	2987228160	2560481280	1234517760	317447424	34012224											9474296896
5	13492928512	60718178304	117099343872	125463582720	80655160320	31109847552	6666395904	612220032										435817657216
6	377801998336	1942981705728	4371708837888	5620768505856	4516688977920	2322868617216	746636341248	137137287168	11019960576									20047612231936
7	10578455953408	61203923730432	157381518163968	236072277245952	227641124487168	146340722884608	62717452664832	17279298183168	2777030065152	198359290368								922190162669056
8	296196766695424	1904122071613440	5508353135738880	9442891089838080	10623252476067840	8195080481538048	4390221686538240	1612734497095680	388784209121280	55540601303040	3570467226624							42420747482776576
9	8293509467471872	58646959805693952	188508085089730560	363551306958766080	467423108946984960	420680798052286464	270437655890755584	124180556276367360	39915178803118080	8553252600668160	1099703905800192	64268410079232						1951354384207722496
10	232218265089212416	1791398044973924352	6333871659014946816	13572582126460600320	19631770575773368320	20192678306509750272	15144508729882312704	8344933381771886592	3352875019461918720	957964291274833920	184750256174432256	21594185786621952	1156831381426176					89762301673555234816
11	6502111422497947648	54339074030875705344	209593571261949149184	494041989403165851648	793996054397945118720	918766862946193637376	787514453953880260608	506259291827494453248	244089301416827682816	87174750506009886720	22416364415831113728	3930141813165195264	421086622839128064	20822964865671168				4129065876983540801536
12	182059119829942534144	1638532078469482807296	6846723327890338873344	17605859986003728531456	31124645332399448653824	40017401141656433983488	38588208243740132769792	28350520342339689381888	15947167692566075277312	6834500439671175118848	2196803712751449145344	513538530253585514496	82532978076469100544	8162602227343097856	374813367582081024			189937030341242876870656
13	5097655355238390956032	49155962354084484218880	221201830593380178984960	616205099510130498600960	1188395549055251675873280	1680730847949570227306496	1800783051374539529256960	1488402317972833692549120	956830061553964516638720	478415030776982258319360	184531511871121728208896	53921545676626479022080	11554616930705674076160	1714146467742050549760	157421614384474030080	6746640616477458432		8737103395697172336050176
14	142734349946674946768896	1468124742308656595337216	7078458578988165727518720	21235375736964497182556160	44366767164729395899269120	68451583625582496530300928	80675080701579370910711808	74089359827981054918000640	53582483447022012931768320	30618561969726864532439040	13778352886377089039597568	4831370492625732520378368	1294117096239035496529920	255979205849479548764160	35262441622122182737920	3022494996181901377536	121439531096594251776	401906756202069927458308096

Technical details

Node id: n-8c08995d48487910 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-9111d60e4a92b32c

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:              784,              1008,                324;
  1:            21952,             42336,              27216,               5832;
  2:           614656,           1580544,            1524096,             653184,             104976;
  3:         17210368,          55319040,           71124480,           45722880,           14696640,            1889568;
  4:        481890304,        1858719744,         2987228160,         2560481280,         1234517760,          317447424,           34012224;
  5:      13492928512,       60718178304,       117099343872,       125463582720,        80655160320,        31109847552,         6666395904,          612220032;
  6:     377801998336,     1942981705728,      4371708837888,      5620768505856,      4516688977920,      2322868617216,       746636341248,       137137287168,       11019960576;
  7:   10578455953408,    61203923730432,    157381518163968,    236072277245952,    227641124487168,    146340722884608,     62717452664832,     17279298183168,     2777030065152,     198359290368;
  8:  296196766695424,  1904122071613440,   5508353135738880,   9442891089838080,  10623252476067840,   8195080481538048,   4390221686538240,   1612734497095680,   388784209121280,   55540601303040,    3570467226624;
  9: 8293509467471872, 58646959805693952, 188508085089730560, 363551306958766080, 467423108946984960, 420680798052286464, 270437655890755584, 124180556276367360, 39915178803118080, 8553252600668160, 1099703905800192, 64268410079232;

%F Formula

The row polynomials satisfy P(n,x) = (28+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 1008*x + 324*x^2.

%t Mathematica

P[0, x_] := 784 + 1008*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 18*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] = 784 + 1008*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 18*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 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 whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.